In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and related branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 points ...
''X'' that satisfies Axiom T
4: 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 cl ...
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 T
4 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 T
5 spaces, and perfectly normal Hausdorff spaces, or T
6 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 points ...
''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 cl ...
s ''E'' and ''F'', there are
neighbourhoods
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
''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 T
4 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 set
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 ...
s 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 T
5 space, or completely T
4 space, is a completely normal T
1 space ''X'', which implies that ''X'' is Hausdorff; equivalently, every subspace of ''X'' must be a T
4 space.
A perfectly normal space is a topological space
in which every two disjoint closed sets
and
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 a ...
, in the sense that there is a continuous function
from
to the interval