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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
and related branches of mathematics, a normal space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' that satisfies Axiom T4: every two disjoint
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s of ''X'' have disjoint
open neighborhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
s. A normal
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
is also called a T4 space. These conditions are examples of
separation axiom In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...
s and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.


Definitions

A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is a normal space if, given any disjoint
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s ''E'' and ''F'', there are
neighbourhoods A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
''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 In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. 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'' with subspace topology 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 topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets ...
, in the sense that there is a continuous function f from X to the interval ,1/math> such that f^(0)=E and f^(1)=F. (This is a stronger separation property than normality, as by
Urysohn's lemma In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Section 15. Urysohn's lemma is commonly used to construct continu ...
disjoint closed sets in a normal space can be
separated by a function In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets ...
, in the sense of E\subseteq f^(0) and F\subseteq f^(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 a
zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equ ...
. 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 In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of ''subobject'' depends on the context. These properties are particularly considered in topology and graph theory, but al ...
. 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 The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, th ...
. Terms like "normal
regular space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can b ...
" 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 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 norma ...
s and fully T4 spaces are discussed elsewhere; they are related to
paracompactness 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 norma ...
. A
locally normal space In mathematics, particularly topology, a topological space ''X'' is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighb ...
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 In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. ...
.


Examples of normal spaces

Most spaces encountered in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
are normal Hausdorff spaces, or at least normal regular spaces: * All
metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
(and hence all
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
s) are perfectly normal Hausdorff; * All
pseudometric space In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metr ...
s (and hence all
pseudometrisable space In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metr ...
s) are perfectly normal regular, although not in general Hausdorff; * All
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
Hausdorff spaces are normal; * In particular, the
Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone ...
of a
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
is normal Hausdorff; * Generalizing the above examples, all
paracompact 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 norma ...
Hausdorff spaces are normal, and all paracompact regular spaces are normal; * All paracompact
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
s are perfectly normal Hausdorff. However, there exist non-paracompact manifolds that are not even normal. * All order topologies on
totally ordered set In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...
s are hereditarily normal and Hausdorff. * Every regular
second-countable space In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
is completely normal, and every regular
Lindelöf space In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of '' compactness'', which requires the existence of a ''finite'' s ...
is normal. Also, all
fully normal 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 norma ...
s are normal (even if not regular).
Sierpiński space In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named ...
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 In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
or on the
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
, which is used in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
. A non-normal space of some relevance to analysis is the
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
of all functions from the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
R to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of uncountably many non-
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued functions, as expressed by the following theorems valid for any normal space ''X''.
Urysohn's lemma In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Section 15. Urysohn's lemma is commonly used to construct continu ...
: 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 In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
,1 (In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also
separated by a function In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets ...
.) More generally, the
Tietze extension theorem In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundednes ...
: 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 In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly giv ...
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 In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
of a normal space ''X'', then there is a
partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are ...
precisely subordinate to U. (This shows the relationship of normal spaces to
paracompactness 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 norma ...
.) In fact, any space that satisfies any one of these three conditions must be normal. A
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
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 topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line \mathbb under the half-open inter ...
. In fact, since there exist spaces which are Dowker, a product of a normal space and
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
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 In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces ,\omega_1/math> and ,\ ...
. 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 In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is tran ...
) 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 In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
. Thus, anything from "normal R0" to "normal completely regular" is the same as what we usually call ''normal regular''. Taking
Kolmogorov quotient In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing th ...
s, 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 In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named ...
is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.


See also

* *


Citations


References

* Engelking, Ryszard, ''General Topology'', Heldermann Verlag Berlin, 1989. * * * * *{{cite book , last= Willard , first= Stephen , title= General Topology , publisher= Addison-Wesley , location= Reading, MA , year= 1970 , isbn= 978-0-486-43479-7 , url= https://archive.org/details/generaltopology00will_0 , url-access= registration Properties of topological spaces Separation axioms