# 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 $T_{0}$ $T_{1}$ $T_{2}$ $T_{2\frac{1}{2}}$ $T_{3}$ and $T_{0}$ $T_{3\frac{1}{2}}$ and $T_{0}$ $T_{4}$ and $T_{1}$ $T_{4}$ and every closed set is a $G_{\delta}$ (see here (http://planetmath.org/G_deltaSet)) $T_{1}$ and perfectly $T_{4}$ $T_{5}$ 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 $T_{4}$