separation axioms

The separation axioms are additional conditions which may be required to a topological space in order to ensure that some particular types of sets can be separated by open sets, thus avoiding certain pathological cases.

Axiom	Definition
$T_{0}$	given two distinct points, there is an open set containing exactly one of them;
$T_{1}$ (http://planetmath.org/T1Space)	given two distinct points, there is a neighborhood of each of them which does not contain the other point;
$T_{2}$ (http://planetmath.org/T2Space)	given two distinct points, there are two disjoint open sets each of which contains one of the points;
$T_{2\frac{1}{2}}$	given two distinct points, there are two open sets, each of which contains one of the points, whose closures are disjoint;
$T_{3}$ (http://planetmath.org/T3Space)	given a closed set $A$ and a point $x\notin A$ , there are two disjoint open sets $U$ and $V$ such that $x\in U$ and $A\subset V$ ;
$T_{3\frac{1}{2}}$	given a closed set $A$ and a point $x\notin A$ , there is an Urysohn function for $A$ and $\{b\}$ ;
$T_{4}$	given two disjoint closed sets $A$ and $B$ , there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$ ;
$T_{5}$	given two separated sets $A$ and $B$ , there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$ .

If a topological space satisfies a $T_{i}$ axiom, it is called a $T_{i}$ -space. The following table shows other common names for topological spaces with these or other additional separation properties.

Name	Separation properties
Kolmogorov space	$T_{0}$
Fréchet space	$T_{1}$
Hausdorff space	$T_{2}$
Completely Hausdorff space	$T_{2\frac{1}{2}}$
Regular space	$T_{3}$ and $T_{0}$
Tychonoff or completely regular space	$T_{3\frac{1}{2}}$ and $T_{0}$
Normal space	$T_{4}$ and $T_{1}$
Perfectly $T_{4}$ space	$T_{4}$ and every closed set is a $G_{\delta}$ (see here (http://planetmath.org/G_deltaSet))
Perfectly normal space	$T_{1}$ and perfectly $T_{4}$
Completely normal space	$T_{5}$ and $T_{1}$

The following implications hold strictly:

	$\displaystyle(T_{2}\textnormal{ and }T_{3})$	$\displaystyle\Rightarrow T_{2\frac{1}{2}}$
	$\displaystyle(T_{3}\textnormal{ and }T_{4})$	$\displaystyle\Rightarrow T_{3\frac{1}{2}}$
	$\displaystyle T_{3\frac{1}{2}}$	$\displaystyle\Rightarrow T_{3}$
	$\displaystyle T_{5}$	$\displaystyle\Rightarrow T_{4}$

	Completely normal	$\displaystyle\Rightarrow\textnormal{ normal }\Rightarrow\textnormal{ % completely regular }$
		$\displaystyle\Rightarrow\textnormal{ regular }\Rightarrow T_{2\frac{1}{2}}% \Rightarrow T_{2}\Rightarrow T_{1}\Rightarrow T_{0}$

Remark. Some authors define $T_{3}$ spaces in the way we defined regular spaces, and $T_{4}$ spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue.

Bibliography: Counterexamples in Topology, L. A. Steen, J. A. Seebach Jr., Dover Publications Inc. (New York)

Title	separation axioms
Canonical name	SeparationAxioms
Date of creation	2013-03-22 13:28:47
Last modified on	2013-03-22 13:28:47
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	26
Author	Koro (127)
Entry type	Definition
Classification	msc 54D10
Classification	msc 54D15
Synonym	separation properties
Related topic	NormalTopologicalSpace
Related topic	HausdorffSpaceNotCompletelyHausdorff
Related topic	SierpinskiSpace
Related topic	MetricSpacesAreHausdorff
Related topic	ZeroDimensional
Related topic	T2Space
Related topic	RegularSpace
Related topic	T4Space
Defines	Hausdorff
Defines	completely Hausdorff
Defines	normal
Defines	completely normal
Defines	regular
Defines	Tychonoff
Defines	completely regular
Defines	perfectly normal
Defines	Tychonov
Defines	perfectly $T_{4}$