text:
Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as $$\mathbf{Z}[i]$$ or $$\Z[i].$$

Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic.

Gaussian integers are algebraic integers and form the simplest ring of quadratic integers.

Gaussian integers are named after the German mathematician Carl Friedrich Gauss.

Basic definitions
The Gaussian integers are the set
 * $$\mathbf{Z}[i]=\{a+bi \mid a,b\in \mathbf{Z} \}, \qquad \text{ where } i^2 = -1.$$

In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers. It is thus an integral domain.

When considered within the complex plane, the Gaussian integers constitute the $2$-dimensional integer lattice.

The conjugate of a Gaussian integer $a + bi$ is the Gaussian integer $a – bi$.

The norm of a Gaussian integer is its product with its conjugate.
 * $$N(a+bi) = (a+bi)(a-bi) = a^2+b^2.$$

The norm of a Gaussian integer is thus the square of its absolute value as a complex number. The norm of a Gaussian integer is a nonnegative integer, which is a sum of two squares. Thus a norm cannot be of the form $4k + 3$, with $k$ integer.

The norm is multiplicative, that is, one has
 * $$N(zw) = N(z)N(w),$$

for every pair of Gaussian integers $z, w$. This can be shown directly, or by using the multiplicative property of the modulus of complex numbers.

The units of the ring of Gaussian integers (that is the Gaussian integers whose multiplicative inverse is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, –1, $i$ and $–i$.

Euclidean division

Gaussian integers have a Euclidean division (division with remainder) similar to that of integers and polynomials. This makes the Gaussian integers a Euclidean domain, and implies that Gaussian integers share with integers and polynomials many important properties such as the existence of a Euclidean algorithm for computing greatest common divisors, Bézout's identity, the principal ideal property, Euclid's lemma, the unique factorization theorem, and the Chinese remainder theorem, all of which can be proved using only Euclidean division.

A Euclidean division algorithm takes, in the ring of Gaussian integers, a dividend $a$ and divisor $b ≠ 0$, and produces a quotient $q$ and remainder $r$ such that
 * $$a=bq+r\quad \text{and} \quad N(r)<N(b).$$

In fact, one may make the remainder smaller:
 * $$a=bq+r\quad \text{and} \quad N(r)\le \frac{N(b)}{2}.$$

Even with this better inequality, the quotient and the remainder are not necessarily unique, but one may refine the choice to ensure uniqueness.

To prove this, one may consider the complex number quotient $x + iy = a⁄b$. There are unique integers $m$ and $n$ such that $–1⁄2 < x – m ≤ 1⁄2$ and $–1⁄2 < y – n ≤ 1⁄2$, and thus $N(x – m + i(y – n)) ≤ 1⁄2$. Taking $q = m + in$, one has
 * $$a = bq + r,$$

with
 * $$r=b\bigl(x-m+ i(y-n)\bigr), $$

and
 * $$N(r)\le \frac{N(b)}{2}.$$

The choice of $x – m$ and $y – n$ in a semi-open interval is required for uniqueness. This definition of Euclidean division may be interpreted geometrically in the complex plane (see the figure), by remarking that the distance from a complex number $ξ$ to the closest Gaussian integer is at most $√2⁄2$.

Principal ideals
Since the ring $G$ of Gaussian integers is a Euclidean domain, $G$ is a principal ideal domain, which means that every ideal of $G$ is principal. Explicitly, an ideal $I$ is a subset of a ring $R$ such that every sum of elements of $I$ and every product of an element of $I$ by an element of $R$ belong to $I$. An ideal is principal if it consists of all multiples of a single element $g$, that is, it has the form
 * $$\{gx\mid x\in G\}.$$

In this case, one says that the ideal is generated by $g$ or that $g$ is a generator of the ideal.

Every ideal $I$ in the ring of the Gaussian integers is principal, because, if one chooses in $I$ a nonzero element $g$ of minimal norm, for every element $x$ of $I$, the remainder of Euclidean division of $x$ by $g$ belongs also to $I$ and has a norm that is smaller than that of $g$; because of the choice of $g$, this norm is zero, and thus the remainder is also zero. That is, one has $x = qg$, where $q$ is the quotient.

For any $g$, the ideal generated by $g$ is also generated by any associate of $g$, that is, $g, gi, –g, –gi$; no other element generates the same ideal. As all the generators of an ideal have the same norm, the norm of an ideal is the norm of any of its generators.

In some circumstances, it is useful to choose, once for all, a generator for each ideal. There are two classical ways for doing that, both considering first the ideals of odd norm. If the $g = a + bi$ has an odd norm $a^{2} + b^{2}$, then one of $a$ and $b$ is odd, and the other is even. Thus $g$ has exactly one associate with a real part $a$ that is odd and positive. In his original paper, Gauss made another choice, by choosing the unique associate such that the remainder of its division by $2 + 2i$ is one. In fact, as $N(2 + 2i) = 8$, the norm of the remainder is not greater than 4. As this norm is odd, and 3 is not the norm of a Gaussian integer, the norm of the remainder is one, that is, the remainder is a unit. Multiplying $g$ by the inverse of this unit, one finds an associate that has one as a remainder, when divided by $2 + 2i$.

If the norm of $g$ is even, then either $g = 2^{k}h$ or $g = 2^{k}h(1 + i)$, where $k$ is a positive integer, and $N(h)$ is odd. Thus, one chooses the associate of $g$ for getting a $h$ which fits the choice of the associates for elements of odd norm.

Gaussian primes
As the Gaussian integers form a principal ideal domain they form also a unique factorization domain. This implies that a Gaussian integer is irreducible (that is, it is not the product of two non-units) if and only if it is prime (that is, it generates a prime ideal).

The prime elements of $Z[i]$ are also known as Gaussian primes. An associate of a Gaussian prime is also a Gaussian prime. The conjugate of a Gaussian prime is also a Gaussian prime (this implies that Gaussian primes are symmetric about the real and imaginary axes).

A positive integer is a Gaussian prime if and only if it is a prime number that is congruent to 3 modulo 4 (that is, it may be written $4n + 3$, with $n$ a nonnegative integer). The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes.

A Gaussian integer $a + bi$ is a Gaussian prime if and only if either:
 * one of $a, b$ is zero and the absolute value of the other is a prime number of the form $4n + 3$ (with $n$ a nonnegative integer), or
 * both are nonzero and $a^{2} + b^{2}$ is a prime number (which will not be of the form $4n + 3$).

In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, or it is the product of a unit ($±1, ±i$) and a prime number of the form $4n + 3$.

It follows that there are three cases for the factorization of a prime number $p$ in the Gaussian integers:
 * If $p$ is congruent to 3 modulo 4, then it is a Gaussian prime; in the language of algebraic number theory, $p$ is said to be inert in the Gaussian integers.
 * If $p$ is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes (neither is the product of the other by a unit); $p$ is said to be a decomposed prime in the Gaussian integers. For example, $5 = (2 + i)(2 − i)$ and $13 = (3 + 2i)(3 − 2i)$.
 * If $p = 2$, we have $2 = (1 + i)(1 − i) = i(1 − i)^{2}$; that is, 2 is the product of the square of a Gaussian prime by a unit; it is the unique ramified prime in the Gaussian integers.

Unique factorization
As for every unique factorization domain, every Gaussian integer may be factored as a product of a unit and Gaussian primes, and this factorization is unique up to the order of the factors, and the replacement of any prime by any of its associates (together with a corresponding change of the unit factor).

If one chooses, once for all, a fixed Gaussian prime for each equivalence class of associated primes, and if one takes only these selected primes in the factorization, then one obtains a prime factorization which is unique up to the order of the factors. With the choices described above, the resulting unique factorization has the form
 * $$u(1+i)^{e_0}{p_1}^{e_1}\cdots {p_k}^{e_k},$$

where $u$ is a unit (that is, $u ∈ {1, –1, i, –i}$), $e_{0}$ and $k$ are nonnegative integers, $e_{1}, …, e_{k}$ are positive integers, and $p_{1}, …, p_{k}$ are distinct Gaussian primes such that, depending on the choice of selected associates, An advantage of the second choice is that the selected associates behave well under products for Gaussian integers of odd norm. On the other hand, the selected associate for the real Gaussian primes are negative integers. For example, the factorization of 231 in the integers, and with the first choice of associates is 3 × 7 × 11, while it is (–1) × (–3) × (–7) × (–11) with the second choice.
 * either $p_{k} = a_{k} + ib_{k}$ with $a$ odd and positive, and $b$ even,
 * or the remainder of the Euclidean division of $p_{k}$ by $2 + 2i$ equals 1 (this is Gauss's original choice ).

Gaussian rationals
The field of Gaussian rationals is the field of fractions of the ring of Gaussian integers. It consists of the complex numbers whose real and imaginary part are both rational.

The ring of Gaussian integers is the integral closure of the integers in the Gaussian rationals.

This implies that Gaussian integers are quadratic integers and that a Gaussian rational is a Gaussian integer, if and only if it is a solution of an equation
 * $$x^2 +cx+d=0,$$

with $c$ and $d$ integers. In fact $a + bi$ is solution of the equation
 * $$x^2-2ax+a^2+b^2,$$

and this equation has integer coefficients if and only if $a$ and $b$ are both integers.

Greatest common divisor
As for any unique factorization domain, a greatest common divisor (gcd) of two Gaussian integers $a, b$ is a Gaussian integer $d$ that is a common divisor of $a$ and $b$, which has all common divisors of $a$ and $b$ as divisor. That is (where $|$ denotes the divisibility relation),

Thus, greatest is meant relatively to the divisibility relation, and not for an ordering of the ring (for integers, both meanings of greatest coincide).
 * $d | a$ and $d | b$, and
 * $c | a$ and $c | b$ implies $c | d$.

More technically, a greatest common divisor of $a$ and $b$ is a generator of the ideal generated by $a$ and $b$ (this characterization is valid for principal ideal domains, but not, in general, for unique factorization domains).

The greatest common divisor of two Gaussian integers is not unique, but is defined up to the multiplication by a unit. That is, given a greatest common divisor $d$ of $a$ and $b$, the greatest common divisors of $a$ and $b$ are $d, –d, id$, and $–id$.

There are several ways for computing a greatest common divisor of two Gaussian integers $a$ and $b$. When one knows the prime factorizations of $a$ and $b$,
 * $$a = i^k\prod_m {p_m}^{\nu_m}, \quad b = i^n\prod_m {p_m}^{\mu_m},$$

where the primes $p_{m}$ are pairwise non associated, and the exponents $μ_{m}$ non-associated, a greatest common divisor is
 * $$\prod_m {p_m}^{\lambda_m},$$

with
 * $$\lambda_m = \min(\nu_m, \mu_m).$$

Unfortunately, except in simple cases, the prime factorization is difficult to compute, and Euclidean algorithm leads to a much easier (and faster) computation. This algorithm consists of replacing of the input $(a, b)$ by $(b, r)$, where $r$ is the remainder of the Euclidean division of $a$ by $b$, and repeating this operation until getting a zero remainder, that is a pair $(d, 0)$. This process terminates, because, at each step, the norm of the second Gaussian integer decreases. The resulting $d$ is a greatest common divisor, because (at each step) $b$ and $r = a – bq$ have the same divisors as $a$ and $b$, and thus the same greatest common divisor.

This method of computation works always, but is not as simple as for integers because Euclidean division is more complicated. Therefore, a third method is often preferred for hand-written computations. It consists in remarking that the norm $N(d)$ of the greatest common divisor of $a$ and $b$ is a common divisor of $N(a)$, $N(b)$, and $N(a + b)$. When the greatest common divisor $D$ of these three integers has few factors, then it is easy to test, for common divisor, all Gaussian integers with a norm dividing $D$.

For example, if $a = 5 + 3i$, and $b = 2 – 8i$, one has $N(a) = 34$, $N(b) = 68$, and $N(a + b) = 74$. As the greatest common divisor of the three norms is 2, the greatest common divisor of $a$ and $b$ has 1 or 2 as a norm. As a gaussian integer of norm 2 is necessary associated to $1 + i$, and as $1 + i$ divides $a$ and $b$, then the greatest common divisor is $1 + i$.

If $b$ is replaced by its conjugate $b = 2 + 8i$, then the greatest common divisor of the three norms is 34, the norm of $a$, thus one may guess that the greatest common divisor is $a$, that is, that $a | b$. In fact, one has $2 + 8i = (5 + 3i)(1 + i)$.

Congruences and residue classes
Given a Gaussian integer $z_{0}$, called a modulus, two Gaussian integers $z_{1},z_{2}$ are congruent modulo $z_{0}$, if their difference is a multiple of $z_{0}$, that is if there exists a Gaussian integer $q$ such that $z_{1} − z_{2} = qz_{0}$. In other words, two Gaussian integers are congruent modulo $z_{0}$, if their difference belongs to the ideal generated by $z_{0}$. This is denoted as $z_{1} ≡ z_{2} (mod z_{0})$.

The congruence modulo $z_{0}$ is an equivalence relation (also called a congruence relation), which defines a partition of the Gaussian integers into equivalence classes, called here congruence classes or residue classes. The set of the residue classes is usually denoted $Z[i]/z_{0}Z[i]$, or $Z[i]/⟨z_{0}⟩$, or simply $Z[i]/z_{0}$.

The residue class of a Gaussian integer $a$ is the set
 * $$ \bar a := \left\{ z \in \mathbf{Z}[i] \mid z \equiv a \pmod{z_0} \right\}$$

of all Gaussian integers that are congruent to $a$. It follows that $\overline{a} = \overline{b}$ if and only if $a ≡ b (mod z_{0})$.

Addition and multiplication are compatible with congruences. This means that $a_{1} ≡ b_{1} (mod z_{0})$ and $a_{2} ≡ b_{2} (mod z_{0})$ imply $a_{1} + a_{2} ≡ b_{1} + b_{2} (mod z_{0})$ and $a_{1}a_{2} ≡ b_{1}b_{2} (mod z_{0})$. This defines well-defined operations (that is independent of the choice of representatives) on the residue classes:
 * $$\bar a + \bar b := \overline{a+b}\quad \text{and}\quad \bar a \cdot\bar b := \overline{ab}.$$

With these operations, the residue classes form a commutative ring, the quotient ring of the Gaussian integers by the ideal generated by $z_{0}$, which is also traditionally called the residue class ring modulo $z_{0}$ (for more details, see Quotient ring).

Examples

 * There are exactly two residue classes for the modulus $1 + i$, namely $\overline{0} = {0, ±2, ±4,…,±1 ± i, ±3 ± i,…}$ (all multiples of $1 + i$), and $\overline{1} = {±1, ±3, ±5,…, ±i, ±2 ± i,…}$, which form a checkerboard pattern in the complex plane. These two classes form thus a ring with two elements, which is, in fact, a field, the unique (up to an isomorphism) field with two elements, and may thus be identified with the integers modulo 2. These two classes may be considered as a generalization of the partition of integers into even and odd integers. Thus one may speak of even and odd Gaussian integers (Gauss divided further even Gaussian integers into even, that is divisible by 2, and half-even).
 * For the modulus 2 there are four residue classes, namely $\overline{0}, \overline{1}, \overline{i}, \overline{1 + i}$. These form a ring with four elements, in which $x = –x$ for every $x$. Thus this ring is not isomorphic with the ring of integers modulo 4, another ring with four elements. One has $\overline{1 + i}^{2} = \overline{0}$, and thus this ring is not the finite field with four elements, nor the direct product of two copies of the ring of integers modulo 2.
 * For the modulus $2 + 2i = (i − 1)^{3}$ there are eight residue classes, namely $\overline{0}, \overline{±1}, \overline{±i}, \overline{1 ± i}, \overline{2}$, whereof four contain only even Gaussian integers and four contain only odd Gaussian integers.

Describing residue classes

Given a modulus $Q_{00}$, all elements of a residue class have the same remainder for the Euclidean division by $z_{0} = 3 + 2i$, provided one uses the division with unique quotient and remainder, which is described above. Thus enumerating the residue classes is equivalent with enumerating the possible remainders. This can be done geometrically in the following way.

In the complex plane, one may consider a square grid, whose squares are delimited by the two lines
 * $$\begin{align}

V_s&=\left\{ \left. z_0\left(s-\tfrac12 +ix\right) \right\vert x\in \mathbf R\right\} \quad \text{and} \\ H_t&=\left\{ \left. z_0\left(x+i\left(t-\tfrac12\right)\right) \right\vert x\in \mathbf R\right\}, \end{align}$$

with $z = 2 − 4i ≡ −i (mod z_{0})$ and $z_{0}$ integers (blue lines in the figure). These divide the plane in semi-open squares (where $z_{0}$ and $s$ are integers)
 * $$Q_{mn}=\left\{(s+it)z_0 \left\vert s \in \left [m - \tfrac12, m + \tfrac12\right), t \in \left[n - \tfrac12, n + \tfrac12 \right)\right.\right\}.$$

The semi-open intervals that occur in the definition of $t$ have been chosen in order that every complex number belong to exactly one square; that is, the squares $m$ form a partition of the complex plane. One has
 * $$Q_{mn} = (m+in)z_0+Q_{00}=\left\{(m+in)z_0+z\mid z\in Q_{00}\right\}.$$

This implies that every Gaussian integer is congruent modulo $n$ to a unique Gaussian integer in $Q_{mn}$ (the green square in the figure), which its remainder for the division by $Q_{mn}$. In other words, every residue class contains exactly one element in $z_{0}$.

The Gaussian integers in $Q_{00}$ (or in its boundary) are sometimes called minimal residues because their norm are not greater than the norm of any other Gaussian integer in the same residue class (Gauss called them absolutely smallest residues).

From this one can deduce by geometrical considerations, that the number of residue classes modulo a Gaussian integer $z_{0}$ equals its norm $Q_{00}$ (see below for a proof; similarly, for integers, the number of residue classes modulo $Q_{00}$ is its absolute value $z_{0} = a + bi$).

$$

Residue class fields
The residue class ring modulo a Gaussian integer $N(z_{0}) = a^{2} + b^{2}$ is a field if and only if $$z_0$$ is a Gaussian prime.

If $n$ is a decomposed prime or the ramified prime $|n|$ (that is, if its norm $Q_{mn} = (m + in)z_{0} + Q_{00}$ is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, $Q_{mn}$). It is thus isomorphic to the field of the integers modulo $Q_{00}$.

If, on the other hand, $Q_{mn}$ is an inert prime (that is, $N = N(z_{0})$ is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has $n_{g}$ elements, and it is an extension of degree 2 (unique, up to an isomorphism) of the prime field with $A$ elements (the integers modulo $A + Θ(√A)$).

Primitive residue class group and Euler's totient function
Many theorems (and their proofs) for moduli of integers can be directly transferred to moduli of Gaussian integers, if one replaces the absolute value of the modulus by the norm. This holds especially for the primitive residue class group (also called multiplicative group of integers modulo $k × k$) and Euler's totient function. The primitive residue class group of a modulus $Q_{mn}$ is defined as the subset of its residue classes, which contains all residue classes $k^{2}N + O(k√N)$ that are coprime to $k^{2}n_{g} = k^{2}N + Θ(k√N)$, i.e. $n_{g} = N + Θ(√N⁄k)$. Obviously, this system builds a multiplicative group. The number of its elements shall be denoted by $k^{2}$ (analogously to Euler's totient function $k$ for integers $n_{g} = N = N(z_{0})$).

For Gaussian primes it immediately follows that $z_{0}$ and for arbitrary composite Gaussian integers
 * $$z = i^k\prod_m {p_m}^{\nu_m}$$

Euler's product formula can be derived as
 * $$\phi(z) =\prod_{m\, (\nu_m > 0)} \bigl|{p_m}^{\nu_m}\bigr|^2 \left( 1 - \frac 1{|p_m|{}^2} \right) = |z|^2\prod_{p_m|z}\left( 1 - \frac 1{|p_m|{}^2} \right)$$

where the product is to build over all prime divisors $z_{0}$ of $1 + i$ (with $N(z_{0})$). Also the important theorem of Euler can be directly transferred:
 * For all $N(z_{0})$ with $N(z_{0})$, it holds that $z_{0}$.

Historical background
The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence $N(z_{0}) = p^{2}$ to that of $p^{2}$. Similarly, cubic reciprocity relates the solvability of $p$ to that of $p$, and biquadratic (or quartic) reciprocity is a relation between $n$ and $z$. Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers).

In a footnote he notes that the Eisenstein integers are the natural domain for stating and proving results on cubic reciprocity and indicates that similar extensions of the integers are the appropriate domains for studying higher reciprocity laws.

This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory.

Unsolved problems

Most of the unsolved problems are related to distribution of Gaussian primes in the plane.

 * Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.

There are also conjectures and unsolved problems about the Gaussian primes. Two of them are:
 * The real and imaginary axes have the infinite set of Gaussian primes 3, 7, 11, 19, ... and their associates. Are there any other lines that have infinitely many Gaussian primes on them? In particular, are there infinitely many Gaussian primes of the form $\overline{a}$?
 * Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of a uniformly bounded length? This is known as the Gaussian moat problem; it was posed in 1962 by Basil Gordon and remains unsolved.

title: Gaussian integer


text:
Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.

Definitions
A topological space X is a normal space if, given any disjoint closed sets E and F, there are neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.

A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.

A completely normal space, or , is a topological space X such that every subspace of X is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods. Also, X is completely normal if and only if every open subset of X is normal with the subspace topology.

A T5 space, or completely T4 space, is a completely normal T1 space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T4 space.

A perfectly normal space is a topological space $$X$$ in which every two disjoint closed sets $$E$$ and $$F$$ can be precisely separated by a function, in the sense that there is a continuous function $$f$$ from $$X$$ to the interval $$[0,1]$$ such that $$f^{-1}(0)=E$$ and $$f^{-1}(1)=F$$. This is a stronger separation property than normality, as by Urysohn's lemma disjoint closed sets in a normal space can be separated by a function, in the sense of $$E\subseteq f^{-1}(0)$$ and $$F\subseteq f^{-1}(1)$$, but not precisely separated in general. It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X is perfectly normal if and only if every closed set is the zero set of a continuous function. The equivalence between these three characterizations is called Vedenissoff's theorem. Every perfectly normal space is completely normal, because perfect normality is a hereditary property.

A T6 space, or perfectly T4 space, is a perfectly normal Hausdorff space.

Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.

Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".

Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.

A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Nemytskii plane.

Examples of normal spaces
Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:
 * All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
 * All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
 * All compact Hausdorff spaces are normal;
 * In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;
 * Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal;
 * All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompact manifolds that are not even normal.
 * All order topologies on totally ordered sets are hereditarily normal and Hausdorff.
 * Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpiński space is an example of a normal space that is not regular.

Examples of non-normal spaces
An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.

A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the product of uncountably many non-compact metric spaces is never normal.

Properties
Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.

The main significance of normal spaces lies in the fact that they admit "enough" continuous real-valued functions, as expressed by the following theorems valid for any normal space X.

Urysohn's lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from X to the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to be entirely within the unit interval [0,1]. In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also separated by a function.

More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A to R, then there exists a continuous function F: X → R that extends f in the sense that F(x) = f(x) for all x in A.

The map $$\emptyset\rightarrow X$$ has the lifting property with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.

If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U. This shows the relationship of normal spaces to paracompactness.

In fact, any space that satisfies any one of these three conditions must be normal.

A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example of this phenomenon is the Sorgenfrey plane. In fact, since there exist spaces which are Dowker, a product of a normal space and [0, 1] need not to be normal. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (Tychonoff's theorem) and the T2 axiom are preserved under arbitrary products.

Relationships to other separation axioms
If a normal space is R0, then it is in fact completely regular. Thus, anything from "normal R0" to "normal completely regular" is the same as what we usually call normal regular. Taking Kolmogorov quotients, we see that all normal T1 spaces are Tychonoff. These are what we usually call normal Hausdorff spaces.

A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.

Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpiński space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Normal space


text:
2014

2014 was designated as:
 * International Year of Crystallography
 * International Year of Family Farming
 * International Year of Small Island Developing States
 * International Year of Solidarity with the Palestinian People

January

 * January 1 – Latvia adopts the euro as its currency and becomes the 18th member of the Eurozone.
 * January 5 – A launch of the communication satellite GSAT-14 aboard the GSLV Mk.II D5 marks the first successful flight of an Indian cryogenic rocket engine.
 * January 9 – The Elk River Chemical Spill in Charleston, West Virginia, when a leakage of the chemical MCMH from a Freedom Industries chemical plant left 300,000 residents of Charleston and nine surrounding counties without drinkable water for over three months.

February

 * February – The West African Ebola virus epidemic begins, infecting at least 28,616 people and killing at least 11,310 people, the most severe both in terms of numbers of infections and casualties.
 * February 7–23 – The XXII Olympic Winter Games are held in Sochi, Russia. Slopestyle events are introduced for the first time.
 * February 13 – Belgium becomes the first country in the world to legalise euthanasia for terminally ill patients of any age.
 * February 22 – Revolution of Dignity: The Verkhovna Rada (Ukrainian parliament) votes to remove President Viktor Yanukovych from office, replacing him with Oleksandr Turchynov, after days of civil unrest leaving around 100 people dead in Kyiv.
 * February 28 – The number of people in the U.S. using mobile devices to access the internet overtook those using desktop computers for the first time, a feat which would be followed globally two years later in 2016.

March

 * March 1 – 2014 Kunming attack: Eight knife-wielding terrorists stabbed and slashed passengers at the Kunming railway station in Kunming, Yunnan, China, killing 31 people and injuring 143 others.
 * March 5 – Nicolás Maduro, the President of Venezuela, severs diplomatic and political ties with Panama, accusing Panama of being involved in a conspiracy against the Venezuelan government.
 * March 8 – Malaysia Airlines Flight 370, a Boeing 777 airliner en route to Beijing from Kuala Lumpur, disappears over the Gulf of Thailand with 239 people on board. The aircraft is presumed to have crashed into the Indian Ocean.
 * March 16
 * 2014 Crimean status referendum: A disputed referendum on the status of the Crimean Peninsula is held.
 * The fifth ICC T20 World Cup in cricket is held in Bangladesh.
 * March 21 – Russia formally annexes Crimea after President Vladimir Putin signs a bill formalizing the process.
 * March 24 – During an emergency meeting, the United Kingdom, the United States, Italy, Germany, France, Japan, and Canada temporarily suspend Russia from the G8, recognizing Crimea within Ukraine's international borders and rejecting the validity of the 2014 Crimean referendum.
 * March 31 – The United Nations International Court of Justice rules that Japan's Antarctic whaling program is not scientific but commercial and forbids grants of further permits.

April

 * April 7 – The self-proclaimed Donetsk People's Republic statelet unilaterally declares its independence from Ukraine.
 * April 10 – In response to the invasion and subsequent annexation of Crimea by Russia, the Parliamentary Assembly of the Council of Europe (PACE) passes a resolution to temporarily strip Russia of its voting rights; its rights to be represented in the Bureau of the Assembly, the PACE Presidential Committee, and the PACE Standing Committee; and its right to participate in election-observation missions.
 * April 14 – Chibok schoolgirls kidnapping: an estimated 276 girls and women are abducted from a school in Nigeria and held hostage.
 * April 16 – The South Korean ferry MV Sewol capsizes and sinks after an unmanageable cargo shift, killing 304 people, mostly high school students.
 * April 27 – The Catholic Church simultaneously canonizes Popes John XXIII and John Paul II.
 * April 28 – United States President Barack Obama's new economic sanctions against Russia go into effect, targeting companies and individuals close to Russian President Vladimir Putin.

May

 * May 2
 * 2014 Odesa clashes
 * May 5
 * The World Health Organization identifies the spread of poliomyelitis in at least 10 countries as a major worldwide health emergency.
 * Boko Haram militants kill approximately 300 people in a night attack on Gamboru Ngala in Nigeria.
 * May 6–10 – The Eurovision Song Contest 2014 takes place in Copenhagen, Denmark, and is won by Austrian entrant Conchita Wurst with the song "Rise Like a Phoenix".
 * May 12 – The self-proclaimed Luhansk People's Republic statelet unilaterally declares its independence from Ukraine.
 * May 14 – According to a South Korea Unification Ministry report, a twenty-three story apartment building collapsed in Pyongchon-guyok, Pyongyang, North Korea (DPR of Korea), with more than 490 fatalities estimated.
 * May 20 – 2014 Jos bombings: Terrorists in Nigeria detonate bombs at Jos, killing 118 people.
 * May 22
 * 2014 Thai coup d'état: The Royal Thai Army overthrows the caretaker government of Niwatthamrong Boonsongpaisan after a failure to resolve the political unrest in Thailand.
 * A terrorist attack in Ürümqi, China leaves 43 dead and more than 90 injured.
 * The Donetsk People's Republic and the Luhansk People's Republic declare the formation of Novorossiya, also referred to as the Union of People's Republics.
 * May 26 – Narendra Modi succeeds Manmohan Singh as the 14th prime minister of India.

June

 * June 5 – A Sunni militant group now calling itself the Islamic State of Iraq and the Levant (also known as ISIS, ISIL or Daesh) begins an offensive through northern Iraq, aiming to capture the Iraqi capital city of Baghdad and overthrow the Shiite government led by Prime Minister Nouri al-Maliki.
 * June 12 – In the Camp Speicher massacre in Iraq, ISIL kills 1,566 Shia Iraqi Air Force cadets. It is the second deadliest terrorist attack in history and the deadliest attack conducted by ISIL.
 * June 13 – International military intervention against ISIL begins.
 * June 14 – The Catholic Church removes restrictions on clerical marriage in the Eastern Catholic Churches' diaspora.
 * June 15
 * The San Antonio Spurs defeat the Miami Heat in the 2014 NBA Finals, 4–1, to win their 5th championship in franchise history.
 * Australia wins the 2014 Hockey World Cup.
 * June 19 – Felipe VI becomes King of Spain upon the abdication of his father, Juan Carlos I.
 * June 29 – The Islamic State of Iraq and the Levant declares itself a caliphate.

July

 * July 8 – Brazil, hosts of the 2014 FIFA World Cup, are beaten 7–1 by Germany in the semi-finals.
 * July 8–August 26 – Amid growing tensions between Israel and Hamas following the kidnapping and murder of three Israeli teenagers in June and the revenge killing of a Palestinian teenager in July, Israel launches Operation Protective Edge against Hamas-controlled Gaza Strip starting with numerous missile strikes, followed by a ground offensive a week later. In seven weeks of fighting, 2,100 Palestinians and 71 Israelis are killed.
 * July 17 – Malaysia Airlines Flight 17, a Boeing 777, crashes in eastern Ukraine after being shot down by a missile. All 298 people on board are killed.
 * July 18 – Typhoon Rammasun makes landfall in Hainan, China as a category 5 super typhoon, killing 88 and causing $7.14 billion in damages.
 * July 20 – Monty Python Live (Mostly) closes in London, ending the famous British comedy group Monty Python, active since 1969.
 * July 21 – The United Nations Security Council adopts Resolution 2166 in response to the shootdown of Malaysia Airlines Flight 17.
 * July 23–August 3 – The 2014 Commonwealth Games takes place in Glasgow, Scotland.
 * July 24 – Air Algérie Flight 5017, a McDonnell Douglas MD-83, crashes in Mali, killing all 116 people on board.
 * July 28 – 100-year anniversary of World War I's commencement (Austria-Hungary declares war on Serbia).

August

 * August 3 – Sinjar massacre, beginning of ISIS attacks resulting in massacre of over 4,000 Yazidis in Iraq's Sinjar District.
 * August 7 – Khmer Rouge leaders Nuon Chea and Khieu Samphan are found guilty of crimes against humanity and are sentenced to life imprisonment by the Khmer Rouge Tribunal.
 * August 8 – American-led intervention in Iraq: The United States military begins an air campaign in northern Iraq to stem the influx of ISIL militants.
 * August 9 – The shooting of Michael Brown, an African American, by a police officer occurs in Ferguson, Missouri, triggering riots.

September

 * September 15 – Syrian civil war: ISIS militants launched an offensive to capture the town of Kobanî, which lasted for six months.
 * September 18 – In the 2014 Scottish independence referendum, Scotland votes against independence from the United Kingdom.
 * September 21 – The Battle of Sanaa leads to the start of the Yemeni Civil War, as Houthi forces capture the capital city Sanaa, followed by a rapid Houthi takeover of the government.
 * September 22 – American-led intervention in Syria: The United States and several Arab partners begin their airstrike campaign in Syria.
 * September 24 – The Mars Orbiter Mission, popularly known as Mangalyaan, which was launched on 5 November 2013 by the Indian Space Research Organisation, reached Martian orbit, making India the first Asian country to successfully reach Mars.
 * September 26
 * The 2014 World Summit of Nobel Peace Laureates, to be held in Cape Town from 13 to 15 October, is suspended after a boycott of Nobel Laureates to protest the third time refusal of a visa to the 14th Dalai Lama by a South African Government "kowtowing to China".
 * Forty-three male students from the Ayotzinapa Rural Teachers' College are forcibly abducted in Iguala, Guerrero, Mexico. They were allegedly taken into custody by local police officers in collusion with organized crime. The mass kidnapping caused international protests and social unrest, leading to the resignation of Guerrero Governor Ángel Aguirre Rivero.
 * September 27 – Mount Ontake, in Japan, erupts, killing several climbers on the slopes of the volcano.
 * September 28 – Hong Kong protests: Benny Tai Yiu-ting announces that Occupy Central is launched as Hong Kong's government headquarters is being occupied by thousands of protesters. Hong Kong police resort to tear gas to disperse protesters but thousands remain.

October

 * October 3 – Stefan Löfven replaces Fredrik Reinfeldt as Prime Minister of Sweden.
 * October 19 – The Roman Catholic Church beatifies Pope Paul VI.
 * October 20 – Joko Widodo is inaugurated as the 7th President of Indonesia.
 * October 24 – Alan Eustace, an American computer scientist, sets a world record highest and longest free fall jump from 135908 ft over Roswell, New Mexico, United States, breaking the sound barrier without any machine assistance during a record space dive out of a massive helium-filled balloon. His descent to Earth lasts 4 minutes 27 seconds and stretches nearly 26 mi with peak speeds exceeding 822 mph, setting new world records for the highest free-fall jump and total free-fall distance 123414 ft.
 * October 31 – Longtime Burkina Faso President Blaise Compaoré resigns after widespread protests in response to his attempts to abolish presidential term limits.

November

 * November 1 – Bangladesh faces a nationwide blackout.
 * November 2 – The Intergovernmental Panel on Climate Change (IPCC) releases the final part of its Fifth Assessment Report, warning that the world faces "severe, pervasive and irreversible" damage from global emissions of CO2.
 * November 3 – The tallest building in the Western Hemisphere, One World Trade Center in New York City, opens.
 * November 12 – The uncrewed Rosetta spacecraft's Philae probe successfully lands on Comet 67P, the first time in history that a spacecraft has landed on such an object.

December

 * December 3 – The Japanese space agency, JAXA, launches uncrewed spaceprobe Hayabusa2 from the Tanegashima Space Center on a six-year round-trip mission to Ryugu to collect rock samples.
 * December 15 – Two hostages and terrorist Man Haron Monis are killed in the Lindt Cafe siege in Sydney, Australia.
 * December 16 – 2014 Peshawar school massacre: The Pakistani Taliban carry out a mass shooting at an army school in Peshawar, Pakistan, killing at least 145 people, mostly schoolchildren.
 * December 17 – U.S. President Barack Obama announces the resumption of normal diplomatic relations between the U.S. and Cuba.
 * December 24 – A fire ignited in the chapel of the Convent of Santa Cruz do Buçaco, in Portugal, destroying A Sagrada Família, a XVII century painting by famous female baroque painter Josefa de Óbidos.
 * December 28 – Indonesia AirAsia Flight 8501 from Surabaya, Indonesia to Singapore crashes into the Java Sea just southwest of Borneo, killing all 162 people on board.
 * December 31 – A stampede occurred on New Years' Eve in Shanghai, China, resulting in the deaths of 36 people and leaving 49 injured.

Nobel Prizes

 * Chemistry – Eric Betzig, Stefan Hell and William E. Moerner
 * Economics – Jean Tirole
 * Literature – Patrick Modiano
 * Peace – Kailash Satyarthi and Malala Yousafzai
 * Physics – Isamu Akasaki, Hiroshi Amano and Shuji Nakamura
 * Physiology or Medicine – John O'Keefe, May-Britt Moser and Edvard Moser

New English words

 * acute flaccid myelitis
 * aro
 * Black Lives Matter
 * gig worker
 * hard pass
 * initial coin offering/ICO
 * manspreading
 * on fleek
 * zoom

title: 2014


text:
Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by. Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.

The notion of paracompact space is also studied in pointless topology, where it is more well-behaved. For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracompact spaces may not be paracompact. Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. However, the product of a paracompact space and a compact space is always paracompact.

Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space.

Definition
A cover of a set $$X$$ is a collection of subsets of $$X$$ whose union contains $$X$$. In symbols, if $$U = \{U_\alpha : \alpha \in A\}$$ is an indexed family of subsets of $$X$$, then $$U$$ is a cover of $$X$$ if

 * $$X \subseteq \bigcup_{\alpha \in A}U_{\alpha}.$$

A cover of a topological space $$X$$ is open if all its members are open sets. A refinement of a cover of a space $$X$$ is a new cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols, the cover $$V = \{V_\beta : \beta \in B\}$$ is a refinement of the cover $$U = \{U_\alpha : \alpha \in A\}$$ if and only if, for every $$V_\beta$$ in $$V$$, there exists some $$U_\alpha$$ in $$U$$ such that $$V_\beta \subseteq U_\alpha$$.

An open cover of a space $$X$$ is locally finite if every point of the space has a neighborhood that intersects only finitely many sets in the cover. In symbols, $$U = \{U_\alpha : \alpha \in A\}$$ is locally finite if and only if, for any $$x$$ in $$X$$, there exists some neighbourhood $$V$$ of $$x$$ such that the set

 * $$\left\{ \alpha \in A : U_{\alpha} \cap V \neq \varnothing \right\}$$

is finite. A topological space $$X$$ is now said to be paracompact if every open cover has a locally finite open refinement.

This definition extends verbatim to locales, with the exception of locally finite: an open cover $$U$$ of $$X$$ is locally finite iff the set of opens $$V$$ that intersect only finitely many opens in $$U$$ also form a cover of $$X$$. Note that an open cover on a topological space is locally finite iff its a locally finite cover of the underlying locale.

Examples

 * Every compact space is paracompact.
 * Every regular Lindelöf space is paracompact. In particular, every locally compact Hausdorff second-countable space is paracompact.
 * The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, nor metrizable.
 * Every CW complex is paracompact.
 * (Theorem of A. H. Stone) Every metric space is paracompact. Early proofs were somewhat involved, but an elementary one was found by M. E. Rudin.  Existing proofs of this require the axiom of choice for the non-separable case.  It has been shown that ZF theory is not sufficient to prove it, even after the weaker axiom of dependent choice is added.

Some examples of spaces that are not paracompact include:

 * The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The long line is locally compact, but not second countable.)
 * Another counterexample is a product of uncountably many copies of an infinite discrete space. Any infinite set carrying the particular point topology is not paracompact; in fact it is not even metacompact.
 * The Prüfer manifold P is a non-paracompact surface. (It is easy to find an uncountable open cover of P with no refinement of any kind.)
 * The bagpipe theorem shows that there are 2ℵ1 topological equivalence classes of non-paracompact surfaces.
 * The Sorgenfrey plane is not paracompact despite being a product of two paracompact spaces.

Properties
Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This can be extended to F-sigma subspaces as well.

 * A regular space is paracompact if every open cover admits a locally finite refinement. (Here, the refinement is not required to be open.) In particular, every regular Lindelöf space is paracompact.
 * (Smirnov metrization theorem) A topological space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable.
 * Michael selection theorem states that lower semicontinuous multifunctions from X into nonempty closed convex subsets of Banach spaces admit continuous selection iff X is paracompact.

Although a product of paracompact spaces need not be paracompact, the following are true:

 * The product of a paracompact space and a compact space is paracompact.
 * The product of a metacompact space and a compact space is metacompact.

Both these results can be proved by the tube lemma which is used in the proof that a product of finitely many compact spaces is compact.

Paracompact Hausdorff spaces
Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.

 * (Theorem of Jean Dieudonné) Every paracompact Hausdorff space is normal.
 * Every paracompact Hausdorff space is a shrinking space, that is, every open cover of a paracompact Hausdorff space has a shrinking: another open cover indexed by the same set such that the closure of every set in the new cover lies inside the corresponding set in the old cover.
 * On paracompact Hausdorff spaces, sheaf cohomology and Čech cohomology are equal.

Partitions of unity
The most important feature of paracompact Hausdorff spaces is that they admit partitions of unity subordinate to any open cover. This means the following: if X is a paracompact Hausdorff space with a given open cover, then there exists a collection of continuous functions on X with values in the unit interval [0, 1] such that:

 * for every function f: X → R from the collection, there is an open set U from the cover such that the support of f is contained in U;
 * for every point x in X, there is a neighborhood V of x such that all but finitely many of the functions in the collection are identically 0 in V and the sum of the nonzero functions is identically 1 in V.

In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any open cover (see below). This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).

Partitions of unity are useful because they often allow one to extend local constructions to the whole space. For instance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold looks like Euclidean space and the integral is well known), and this definition is then extended to the whole space via a partition of unity.

Relationship with compactness
There is a similarity between the definitions of compactness and paracompactness: For paracompactness, "subcover" is replaced by "open refinement" and "finite" by is replaced by "locally finite". Both of these changes are significant: if we take the definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases.

Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.

Comparison of properties with compactness
Paracompactness is similar to compactness in the following respects:

 * Every closed subset of a paracompact space is paracompact.
 * Every paracompact Hausdorff space is normal.

It is different in these respects:

 * A paracompact subset of a Hausdorff space need not be closed. In fact, for metric spaces, all subsets are paracompact.
 * A product of paracompact spaces need not be paracompact. The square of the real line R in the lower limit topology is a classical example for this.

Variations
There are several variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:

A topological space is:

 * metacompact if every open cover has an open point-finite refinement.
 * orthocompact if every open cover has an open refinement such that the intersection of all the open sets about any point in this refinement is open.
 * fully normal if every open cover has an open star refinement, and fully T4 if it is fully normal and T1 (see separation axioms).

The adverb "countably" can be added to any of the adjectives "paracompact", "metacompact", and "fully normal" to make the requirement apply only to countable open covers.

Every paracompact space is metacompact, and every metacompact space is orthocompact.

Definition of relevant terms for the variations

 * Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that contain the point. In symbols, the star of x in U = {Uα : α in A} is

 * $$\mathbf{U}^{*}(x) := \bigcup_{U_{\alpha} \ni x}U_{\alpha}.$$

 * The notation for the star is not standardised in the literature, and this is just one possibility.

 * A star refinement of a cover of a space X is a cover of the same space such that, given any point in the space, the star of the point in the new cover is a subset of some set in the old cover. In symbols, V is a star refinement of U = {Uα : α in A} if for any x in X, there exists a Uα in U such that V*(x) is contained in Uα.
 * A cover of a space X is point-finite (or point finite) if every point of the space belongs to only finitely many sets in the cover. In symbols, U is point finite if for any x in X, the set $$\left\{\alpha \in A : x \in U_{\alpha} \right\}$$ is finite.

As the names imply, a fully normal space is normal and a fully T4 space is T4. Every fully T4 space is paracompact. In fact, for Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T4 space is the same thing as a paracompact Hausdorff space.

Without the Hausdorff property, paracompact spaces are not necessarily fully normal. Any compact space that is not regular provides an example.

A historical note: fully normal spaces were defined before paracompact spaces, in 1940, by John W. Tukey. The proof that all metrizable spaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces full normality and paracompactness are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later Ernest Michael gave a direct proof of the latter fact and M.E. Rudin gave another, elementary, proof.

title: Paracompact space


text:
Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.

In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces.

Formal definition
Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such that $$x\in U\subseteq K$$.

There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But they are not equivalent in general:
 * 1. every point of X has a compact neighbourhood.
 * 2. every point of X has a closed compact neighbourhood.
 * 2′. every point of X has a relatively compact neighbourhood.
 * 2″. every point of X has a local base of relatively compact neighbourhoods.
 * 3. every point of X has a local base of compact neighbourhoods.
 * 4. every point of X has a local base of closed compact neighbourhoods.
 * 5. X is Hausdorff and satisfies any (or equivalently, all) of the previous conditions.

Logical relations among the conditions:
 * Each condition implies (1).
 * Conditions (2), (2′), (2″) are equivalent.
 * Neither of conditions (2), (3) implies the other.
 * Condition (4) implies (2) and (3).
 * Compactness implies conditions (1) and (2), but not (3) or (4).

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact. Spaces satisfying (1) are also called , as they satisfy the weakest of the conditions here.

As they are defined in terms of relatively compact sets, spaces satisfying (2), (2'), (2") can more specifically be called locally relatively compact. Steen & Seebach calls (2), (2'), (2") strongly locally compact to contrast with property (1), which they call locally compact.

Spaces satisfying condition (4) are exactly the  spaces. Indeed, such a space is regular, as every point has a local base of closed neighbourhoods. Conversely, in a regular locally compact space suppose a point $$x$$ has a compact neighbourhood $$K$$. By regularity, given an arbitrary neighbourhood $$U$$ of $$x$$, there is a closed neighbourhood $$V$$ of $$x$$ contained in $$K\cap U$$ and $$V$$ is compact as a closed set in a compact set.

Condition (5) is used, for example, in Bourbaki. Any space that is locally compact (in the sense of condition (1)) and also Hausdorff automatically satisfies all the conditions above. Since in most applications locally compact spaces are also Hausdorff, these locally compact Hausdorff (LCH) spaces will thus be the spaces that this article is primarily concerned with.

Compact Hausdorff spaces
Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space. Here we mention only:
 * the unit interval [0,1];
 * the Cantor set;
 * the Hilbert cube.

Locally compact Hausdorff spaces that are not compact

 * The Euclidean spaces Rn (and in particular the real line R) are locally compact as a consequence of the Heine–Borel theorem.
 * Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact. This even includes nonparacompact manifolds such as the long line.
 * All discrete spaces are locally compact and Hausdorff (they are just the zero-dimensional manifolds). These are compact only if they are finite.
 * All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology. This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version).
 * The space Qp of p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus one point. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

Hausdorff spaces that are not locally compact
As mentioned in the following section, if a Hausdorff space is locally compact, then it is also a Tychonoff space. For this reason, examples of Hausdorff spaces that fail to be locally compact because they are not Tychonoff spaces can be found in the article dedicated to Tychonoff spaces. But there are also examples of Tychonoff spaces that fail to be locally compact, such as:

 * the space Q of rational numbers (endowed with the topology from R), since any neighborhood contains a Cauchy sequence corresponding to an irrational number, which has no convergent subsequence in Q;
 * the subspace $$\{(0, 0)\} \cup ((0, \infty) \times \mathbf{R})$$ of $$\mathbf{R}^2$$, since the origin does not have a compact neighborhood;
 * the lower limit topology or upper limit topology on the set R of real numbers (useful in the study of one-sided limits);
 * any T0, hence Hausdorff, topological vector space that is infinite-dimensional, such as an infinite-dimensional Hilbert space.

The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces in the previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it is finite-dimensional (in which case it is a Euclidean space). This example also contrasts with the Hilbert cube as an example of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point in Hilbert space.

Non-Hausdorff examples

 * The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in senses (3) or (4).
 * The particular point topology on any infinite set is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire space, which is non-compact.
 * The disjoint union of the above two examples is locally compact in sense (1) but not in senses (2), (3) or (4).
 * The right order topology on the real line is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire non-compact space.
 * The Sierpiński space is locally compact in senses (1), (2) and (3), and compact as well, but it is not Hausdorff or regular (or even preregular) so it is not locally compact in senses (4) or (5). The disjoint union of countably many copies of Sierpiński space is a non-compact space which is still locally compact in senses (1), (2) and (3), but not (4) or (5).
 * More generally, the excluded point topology is locally compact in senses (1), (2) and (3), and compact, but not locally compact in senses (4) or (5).
 * The cofinite topology on an infinite set is locally compact in senses (1), (2), and (3), and compact as well, but it is not Hausdorff or regular so it is not locally compact in senses (4) or (5).
 * The indiscrete topology on a set with at least two elements is locally compact in senses (1), (2), (3), and (4), and compact as well, but it is not Hausdorff so it is not locally compact in sense (5).

General classes of examples

 * Every space with an Alexandrov topology is locally compact in senses (1) and (3).

Properties
Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorff space is a Tychonoff space. Since straight regularity is a more familiar condition than either preregularity (which is usually weaker) or complete regularity (which is usually stronger), locally compact preregular spaces are normally referred to in the mathematical literature as locally compact regular spaces. Similarly locally compact Tychonoff spaces are usually just referred to as locally compact Hausdorff spaces.

Every locally compact regular space, in particular every locally compact Hausdorff space, is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every countable union of nowhere dense subsets is empty.

A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is locally closed in Y (that is, X can be written as the set-theoretic difference of two closed subsets of Y). In particular, every closed set and every open set in a locally compact Hausdorff space is locally compact. Also, as a corollary, a dense subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is open in Y. Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be locally closed in Y, although the converse does not hold in general.

Without the Hausdorff hypothesis, some of these results break down with weaker notions of locally compact. Every closed set in a weakly locally compact space (= condition (1) in the definitions above) is weakly locally compact. But not every open set in a weakly locally compact space is weakly locally compact. For example, the one-point compactification $$\Q^*$$ of the rational numbers $$\Q$$ is compact, and hence weakly locally compact. But it contains $$\Q$$ as an open set which is not weakly locally compact.

Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.

For functions defined on a locally compact space, local uniform convergence is the same as compact convergence.

The point at infinity
This section explores compactifications of locally compact spaces. Every compact space is its own compactification. So to avoid trivialities it is assumed below that the space X is not compact.

Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space $$b(X)$$ using the Stone–Čech compactification. But in fact, there is a simpler method available in the locally compact case; the one-point compactification will embed X in a compact Hausdorff space $$a(X)$$ with just one extra point. (The one-point compactification can be applied to other spaces, but $$a(X)$$ will be Hausdorff if and only if X is locally compact and Hausdorff.) The locally compact Hausdorff spaces can thus be characterised as the open subsets of compact Hausdorff spaces.

Intuitively, the extra point in $$a(X)$$ can be thought of as a point at infinity. The point at infinity should be thought of as lying outside every compact subset of X. Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea. For example, a continuous real or complex valued function f with domain X is said to vanish at infinity if, given any positive number e, there is a compact subset K of X such that $$|f(x)| < e$$ whenever the point x lies outside of K. This definition makes sense for any topological space X. If X is locally compact and Hausdorff, such functions are precisely those extendable to a continuous function g on its one-point compactification $$a(X) = X \cup \{ \infty \}$$ where $$g(\infty) = 0.$$

Gelfand representation
For a locally compact Hausdorff space X, the set $$C_0(X)$$ of all continuous complex-valued functions on X that vanish at infinity is a commutative C*-algebra. In fact, every commutative C*-algebra is isomorphic to $$C_0(X)$$ for some unique (up to homeomorphism) locally compact Hausdorff space X. This is shown using the Gelfand representation.

Locally compact groups
The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate measurable functions defined on G. The Lebesgue measure on the real line $$\R$$ is a special case of this.

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. More precisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups. The study of locally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelian locally compact groups.

title: Locally compact space


text:
Pulitzer Prize for Breaking News Photography

The Pulitzer Prize for Breaking News Photography is one of the American Pulitzer Prizes annually awarded for journalism. From 2000 it has used the "breaking news" name but it is considered a continuation of the Pulitzer Prize for Spot News Photography, which was awarded from 1968 to 1999. Prior to 1968, a single Prize was awarded for photojournalism, the Pulitzer Prize for Photography, which was replaced in that year by Pulitzer Prize for Spot News Photography and Pulitzer Prize for Feature Photography.

List of winners for Pulitzer Prize for Spot News Photography
There were 33 Spot News Photography prizes awarded in 32 years including two in 1977 (for 1976 work):
 * 1968: Rocco Morabito, Jacksonville Journal, for his photograph of telephone linemen, "The Kiss of Life".
 * 1969: Edward T. Adams, Associated Press, for his photograph, "Saigon Execution".
 * 1970: Steve Starr, Associated Press, for his news photo taken at Cornell University, "Campus Guns".
 * 1971: John Paul Filo, Valley Daily News/Daily Dispatch, of the Pittsburgh suburbs of Tarentum and New Kensington, for his pictorial coverage of the Kent State University tragedy on May 4, 1970.
 * 1972: Horst Faas and Michel Laurent, Associated Press, for their picture series, "Death in Dacca".
 * 1973: Huynh Cong Ut, Associated Press, for his photograph, "The Terror of War", depicting children in flight from a napalm bombing.
 * 1974: Anthony K. Roberts, a freelance photographer of Beverly Hills, California, for his picture series, "Fatal Hollywood Drama", in which an alleged kidnapper was killed.
 * 1975: Gerald H. Gay, Seattle Times, for his photograph of four exhausted firefighters, "Lull in the Battle".
 * 1976: Stanley Forman, Boston Herald-American, for his sequence of photographs of a fire in Boston, including "Fire Escape Collapse", on July 22, 1975.
 * 1977: Stanley Forman, Boston Herald-American, for his photograph The Soiling of Old Glory, which depicts Joseph Rakes attacking Theodore Landsmark — using an American flag as a lance — during a desegregation busing demonstration at Boston City Hall.
 * 1977: Neal Ulevich, of the Associated Press, for a series of photographs of disorder and brutality in the streets of Bangkok.
 * 1978: John Blair, special assignment photographer for United Press International, for a photograph of a broker held hostage at gunpoint by Tony Kiritsis.
 * 1979: Thomas J. Kelly III, Pottstown Mercury, Pennsylvania, for a series called "Tragedy on Sanatoga Road". It is of a man, Richard G., who was holding his wife and daughter hostage as he was running from the police.
 * 1980: Anonymous, Ettela'at, United Press International, for "Firing Squad in Iran". In 2006, the photographer's identity was revealed to be Jahangir Razmi.
 * 1981: Larry C. Price, Fort Worth Star-Telegram, for his photographs from Liberia.
 * 1982: Ron Edmonds, Associated Press, for his coverage of the Reagan assassination attempt.
 * 1983: Bill Foley, Associated Press, for his series of pictures of victims and survivors of the massacre in the Sabra Camp in Beirut.
 * 1984: Stan Grossfeld, Boston Globe, for his series of photographs which reveal the effects of war on the people of Lebanon.
 * 1985: Photography staff, Register, Santa Ana, California, for their coverage of the Olympic Games.
 * 1986: Carol Guzy and Michel du Cille, Miami Herald, for their photographs of the devastation caused by the eruption of the Nevado del Ruiz volcano in Colombia.
 * 1987: Kim Komenich, San Francisco Examiner, for his photographic coverage of the fall of Ferdinand Marcos.
 * 1988: Scott Shaw, Odessa American, for his photograph of the child Jessica McClure being rescued from the well into which she had fallen.
 * 1989: Ron Olshwanger, a freelance photographer, for a picture published in the St. Louis Post-Dispatch of a firefighter giving mouth-to-mouth resuscitation to a child pulled from a burning building.
 * 1990: Photo staff of the Oakland Tribune, California, for their photographs of devastation caused by the Loma Prieta earthquake of October 17, 1989. The Oakland Tribune team consisted of Tom Duncan, Angela Pancrazio, Pat Greenhouse, Reginald Pearman, Matthew Lee, Gary Reyes, Michael Macor, Ron Riesterer, Paul Miller, Roy H. Williams.
 * 1991: Greg Marinovich, Associated Press, for a series of photographs of supporters of South Africa's African National Congress murdering a man they believed to be a Zulu spy.
 * 1992: Staff, Associated Press, for photographs of the attempted coup in Russia and the subsequent collapse of the Communist regime.
 * 1993: Ken Geiger and William Snyder, Dallas Morning News, for their photographs of the 1992 Summer Olympics in Barcelona.
 * 1994: Paul Watson, Toronto Star, for his photograph, published around the world, of a U.S. soldier's body being dragged by Somalis through the streets of Mogadishu.
 * 1995: Carol Guzy, Washington Post, for her series of photographs illustrating the crisis in Haiti and its aftermath.
 * 1996: Charles Porter IV, a freelancer, for his photographs taken after the Oklahoma City bombing and distributed by the Associated Press, showing a one-year-old victim (Baylee Almon) handed to and then cradled by a fireman.
 * 1997: Annie Wells, Press Democrat, Santa Rosa, California, for her photograph of a firefighter rescuing a teenager from raging floodwaters.
 * 1998: Martha Rial, Pittsburgh Post-Gazette, for her portraits of survivors of the conflicts in Rwanda and Burundi.
 * 1999: Staff, Associated Press, for its portfolio of images following the embassy bombing in Kenya and Tanzania.

List of winners for Pulitzer Prize for Breaking News Photography
One Breaking News Pulitzer has been awarded annually from 2000 without exception.


text:
Nowhere dense set

In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas the interval (0, 1) is not nowhere dense.

A countable union of nowhere dense sets is called a meagre set. Meagre sets play an important role in the formulation of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis.

Definition
Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density: "A subset $S$ of a topological space $X$ is said to be dense in another set $U$ if the intersection $S \cap U$ is a dense subset of $U.$ $S$ is ' or ' in $X$ if $S$ is not dense in any nonempty open subset $U$ of $X.$" Expanding out the negation of density, it is equivalent to require that each nonempty open set $$U$$ contains a nonempty open subset disjoint from $$S.$$ It suffices to check either condition on a base for the topology on $$X.$$  In particular, density nowhere in $$\R$$ is often described as being dense in no open interval.

Definition by closure
The second definition above is equivalent to requiring that the closure, $$\operatorname{cl}_X S,$$ cannot contain any nonempty open set. This is the same as saying that the interior of the closure of $$S$$ is empty; that is,"$\operatorname{int}_X \left(\operatorname{cl}_X S\right) = \varnothing.$"Alternatively, the complement of the closure $$X \setminus \left(\operatorname{cl}_X S\right)$$ must be a dense subset of $$X;$$ in other words, the exterior of $$S$$ is dense in $$X.$$

Properties
The notion of nowhere dense set is always relative to a given surrounding space. Suppose $$A\subseteq Y\subseteq X,$$ where $$Y$$ has the subspace topology induced from $$X.$$ The set $$A$$ may be nowhere dense in $$X,$$ but not nowhere dense in $$Y.$$  Notably, a set is always dense in its own subspace topology. So if $$A$$ is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold:
 * If $$A$$ is nowhere dense in $$Y,$$ then $$A$$ is nowhere dense in $$X.$$
 * If $$Y$$ is open in $$X$$, then $$A$$ is nowhere dense in $$Y$$ if and only if $$A$$ is nowhere dense in $$X.$$
 * If $$Y$$ is dense in $$X$$, then $$A$$ is nowhere dense in $$Y$$ if and only if $$A$$ is nowhere dense in $$X.$$

A set is nowhere dense if and only if its closure is.

Every subset of a nowhere dense set is nowhere dense, and a finite union of nowhere dense sets is nowhere dense. Thus the nowhere dense sets form an ideal of sets, a suitable notion of negligible set. In general they do not form a 𝜎-ideal, as meager sets, which are the countable unions of nowhere dense sets, need not be nowhere dense. For example, the set $$\Q$$ is not nowhere dense in $$\R.$$

The boundary of every open set and of every closed set is closed and nowhere dense. A closed set is nowhere dense if and only if it is equal to its boundary, if and only if it is equal to the boundary of some open set (for example the open set can be taken as the complement of the set). An arbitrary set $$A\subseteq X$$ is nowhere dense if and only if it is a subset of the boundary of some open set (for example the open set can be taken as the exterior of $$A$$).

Examples

 * The set $$S=\{1/n:n=1,2,...\}$$ and its closure $$S\cup\{0\}$$ are nowhere dense in $$\R,$$ since the closure has empty interior.
 * $$\R$$ viewed as the horizontal axis in the Euclidean plane is nowhere dense in $$\R^2.$$
 * $$\Z$$ is nowhere dense in $$\R$$ but the rationals $$\Q$$ are not (they are dense everywhere).
 * $$\Z \cup [(a, b) \cap \Q]$$ is  nowhere dense in $$\R$$: it is dense in the open interval $$(a,b),$$ and in particular the interior of its closure is $$(a,b).$$
 * The empty set is nowhere dense. In a discrete space, the empty set is the  nowhere dense set.
 * In a T1 space, any singleton set that is not an isolated point is nowhere dense.
 * A vector subspace of a topological vector space is either dense or nowhere dense.

Nowhere dense sets with positive measure
A nowhere dense set is not necessarily negligible in every sense. For example, if $$X$$ is the unit interval $$[0, 1],$$ not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure. One such example is the Smith–Volterra–Cantor set.

For another example (a variant of the Cantor set), remove from $$[0, 1]$$ all dyadic fractions, i.e. fractions of the form $$a/2^n$$ in lowest terms for positive integers $$a, n \in \N,$$ and the intervals around them: $$\left(a/2^n - 1/2^{2n+1}, a/2^n + 1/2^{2n+1}\right).$$ Since for each $$n$$ this removes intervals adding up to at most $$1/2^{n+1},$$ the nowhere dense set remaining after all such intervals have been removed has measure of at least $$1/2$$ (in fact just over $$0.535\ldots$$ because of overlaps ) and so in a sense represents the majority of the ambient space $$[0, 1].$$ This set is nowhere dense, as it is closed and has an empty interior: any interval $$(a, b)$$ is not contained in the set since the dyadic fractions in $$(a, b)$$ have been removed.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than $$1,$$ although the measure cannot be exactly 1 (because otherwise the complement of its closure would be a nonempty open set with measure zero, which is impossible).

For another simpler example, if $$U$$ is any dense open subset of $$\R$$ having finite Lebesgue measure then $$\R \setminus U$$ is necessarily a closed subset of $$\R$$ having infinite Lebesgue measure that is also nowhere dense in $$\R$$ (because its topological interior is empty). Such a dense open subset $$U$$ of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers $$\Q$$ is $$0.$$ This may be done by choosing any bijection $$f : \N \to \Q$$ (it actually suffices for $$f : \N \to \Q$$ to merely be a surjection) and for every $$r > 0,$$ letting $$U_r ~:=~ \bigcup_{n \in \N} \left(f(n) - r/2^n, f(n) + r/2^n\right) ~=~ \bigcup_{n \in \N} f(n) + \left(- r/2^n, r/2^n\right)$$ (here, the Minkowski sum notation $$f(n) + \left(- r/2^n, r/2^n\right) := \left(f(n) - r/2^n, f(n) + r/2^n\right)$$ was used to simplify the description of the intervals). The open subset $$U_r$$ is dense in $$\R$$ because this is true of its subset $$\Q$$ and its Lebesgue measure is no greater than $$\sum_{n \in \N} 2 r / 2^n = 2 r.$$ Taking the union of closed, rather than open, intervals produces the F$\sigma$-subset $$S_r ~:=~ \bigcup_{n \in \N} f(n) + \left[- r/2^n, r/2^n\right]$$ that satisfies $$S_{r/2} \subseteq U_r \subseteq S_r \subseteq U_{2r}.$$ Because $$\R \setminus S_r$$ is a subset of the nowhere dense set $$\R \setminus U_r,$$ it is also nowhere dense in $$\R.$$ Because $$\R$$ is a Baire space, the set $$D := \bigcap_{m=1}^{\infty} U_{1/m} = \bigcap_{m=1}^{\infty} S_{1/m}$$ is a dense subset of $$\R$$ (which means that like its subset $$\Q,$$ $$D$$ cannot possibly be nowhere dense in $$\R$$) with $$0$$ Lebesgue measure that is also a nonmeager subset of $$\R$$ (that is, $$D$$ is of the second category in $$\R$$), which makes $$\R \setminus D$$ a comeager subset of $$\R$$ whose interior in $$\R$$ is also empty; however, $$\R \setminus D$$ is nowhere dense in $$\R$$ if and only if its in $$\R$$ has empty interior. The subset $$\Q$$ in this example can be replaced by any countable dense subset of $$\R$$ and furthermore, even the set $$\R$$ can be replaced by $$\R^n$$ for any integer $$n > 0.$$

title: Nowhere dense set


text:
Partition of unity

In mathematics, a partition of unity of a topological space $X$ is a set $R$ of continuous functions from $X$ to the unit interval [0,1] such that for every point $$x\in X$$:
 * there is a neighbourhood of $x$ where all but a finite number of the functions of $R$ are 0, and
 * the sum of all the function values at $x$ is 1, i.e., $\sum_{\rho\in R} \rho(x) = 1.$

Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They are also important in the interpolation of data, in signal processing, and the theory of spline functions.

Existence
The existence of partitions of unity assumes two distinct forms:

 * 1) Given any open cover $$\{ U_i \}_{i \in I}$$ of a space, there exists a partition $$\{ \rho_i \}_{i \in I}$$ indexed over the same set $I$ such that supp $$\rho_i \subseteq U_i.$$ Such a partition is said to be subordinate to the open cover $$\{ U_i \}_i.$$
 * 2) If the space is locally-compact, given any open cover $$\{ U_i \}_{i \in I}$$ of a space, there exists a partition $$\{ \rho_j \}_{j \in J}$$ indexed over a possibly distinct index set $J$ such that each $\rho_j$ has compact support and for each $j \in J$, supp $$\rho_j \subseteq U_i$$ for some $i \in I$.

Thus one chooses either to have the supports indexed by the open cover, or compact supports. If the space is compact, then there exist partitions satisfying both requirements.

A finite open cover always has a continuous partition of unity subordinated to it, provided the space is locally compact and Hausdorff. Paracompactness of the space is a necessary condition to guarantee the existence of a partition of unity subordinate to any open cover. Depending on the category to which the space belongs, it may also be a sufficient condition. The construction uses mollifiers (bump functions), which exist in continuous and smooth manifolds, but not in analytic manifolds. Thus for an open cover of an analytic manifold, an analytic partition of unity subordinate to that open cover generally does not exist. See analytic continuation.

If $R$ and $T$ are partitions of unity for spaces $X$ and $Y$, respectively, then the set of all pairs $$\{ \rho\otimes\tau :\ \rho \in R,\ \tau \in T \}$$ is a partition of unity for the cartesian product space $X \times Y$. The tensor product of functions act as $$(\rho \otimes \tau )(x,y) = \rho(x)\tau(y).$$

Example
We can construct a partition of unity on $$S^1$$ by looking at a chart on the complement of a point $$p \in S^1$$ sending $$S^1 -\{p\}$$ to $$\mathbb{R}$$ with center $$q \in S^1$$. Now, let $$\Phi$$ be a bump function on $$\mathbb{R}$$ defined by $$\Phi(x) = \begin{cases} \exp\left(\frac{1}{x^2-1}\right) & x \in (-1,1) \\ 0 & \text{otherwise} \end{cases}$$ then, both this function and $$1 - \Phi$$ can be extended uniquely onto $$S^1$$ by setting $$\Phi(p) = 0$$. Then, the set $$\{ (S^1 - \{p\}, \Phi), (S^1 - \{q\}, 1-\Phi) \}$$ forms a partition of unity over $$S^1$$.

Variant definitions
Sometimes a less restrictive definition is used: the sum of all the function values at a particular point is only required to be positive, rather than 1, for each point in the space. However, given such a set of functions $$\{ \psi_i \}_{i=1}^\infty$$ one can obtain a partition of unity in the strict sense by dividing by the sum; the partition becomes $$\{ \sigma^{-1}\psi_i \}_{i=1}^\infty$$ where $\sigma(x) := \sum_{i=1}^\infty \psi_i(x)$, which is well defined since at each point only a finite number of terms are nonzero. Even further, some authors drop the requirement that the supports be locally finite, requiring only that $\sum_{i = 1}^\infty \psi_i(x) < \infty$ for all $$x$$.

In the field of operator algebras, a partition of unity is composed of projections $$p_i=p_i^*=p_i^2$$. In the case of $\mathrm{C}^*$-algebras, it can be shown that the entries are pairwise-orthogonal: $$p_ip_j=\delta_{i,j}p_i\qquad (p_i,\,p_j\in R).$$ Note it is not the case that in a general *-algebra that the entries of a partition of unity are pairwise-orthogonal. If $$a$$ is a normal element of a unital $$\mathrm{C}^*$$-algebra $$A$$, and has finite spectrum $$\sigma(a)=\{\lambda_1,\dots,\lambda_N\}$$, then the projections in the spectral decomposition: $$a=\sum_{i=1}^N\lambda_i\,P_i,$$ form a partition of unity.

In the field of compact quantum groups, the rows and columns of the fundamental representation $$u\in M_N(C)$$ of a quantum permutation group $$(C,u)$$ form partitions of unity.

Applications
A partition of unity can be used to define the integral (with respect to a volume form) of a function defined over a manifold: one first defines the integral of a function whose support is contained in a single coordinate patch of the manifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows that the definition is independent of the chosen partition of unity.

A partition of unity can be used to show the existence of a Riemannian metric on an arbitrary manifold.

Method of steepest descent employs a partition of unity to construct asymptotics of integrals.

Linkwitz–Riley filter is an example of practical implementation of partition of unity to separate input signal into two output signals containing only high- or low-frequency components.

The Bernstein polynomials of a fixed degree m are a family of m+1 linearly independent polynomials that are a partition of unity for the unit interval $$[0,1]$$.

Partitions of unity are used to establish global smooth approximations for Sobolev functions in bounded domains.

title: Partition of unity