# Separation axiom

A condition imposed on a topological space, expressing the requirement that some disjoint (i.e. not having common points) sets can be topologically separated from each other in a specific way. The simplest (i.e. weakest) of these axioms apply only to one-point sets, i.e. to the points of a space. These are the so-called axioms $T_0$ (Kolmogorov's separation axiom, cf. also Kolmogorov space; Kolmogorov axiom) and $T_1$. The next in line are $T_2$ (Hausdorff's separation axiom), $T_3$ (regularity axiom) and $T_4$ (normality axiom), which require, respectively, that every two different points (axiom $T_2$), every point and every closed set not containing it (axiom $T_3$), and every two disjoint closed sets (axiom $T_4$) can be separated by neighbourhoods, i.e. are contained in disjoint open sets of the given space.

A topological space which satisfies the axiom $T_i$, $i=2,3,4$, is called a $T_i$-space; a $T_2$-space is also called a Hausdorff space (cf. Hausdorff space), and a $T_3$-space is called regular (cf. Regular space); a Hausdorff $T_4$-space is always regular and is called normal (cf. Normal space).

Functional separation is of particular significance. Two sets $A$ and $B$ in a given topological space $X$ are said to be functionally separated (or completely separated) in $X$ if there exists a real-valued bounded continuous function $f$, defined throughout the space, which takes one value $a$ at all points of the set $A$, and a value $b$, different from $a$, at all points of the set $B$. It can always be supposed that $a=0$, $b=1$, and that $0\leq f(x)\leq1$ at all points $x\in X$.

Two functionally separable sets are always separable by neighbourhoods, but the converse is not always true. However, Urysohn's lemma holds: In a normal space, every two disjoint closed sets are functionally separable. A space in which every point is functionally separable from every closed set not containing it is called completely regular (cf. Completely regular space). A completely regular $T_2$-space is called a Tikhonov space.

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How to Cite This Entry:
Separation axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separation_axiom&oldid=42527
This article was adapted from an original article by V.I. Zaitsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article