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.

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)

Comments

The reader is warned that there is not really one convention here. There are authors who equate $T_3$ and regularity, and $T_4$ and normality and take both to include the $T_1$-property, e.g., [a1].

In [a2] one finds the convention that "T3=regular+T1" and "T4=normal+T1" , where [a3] adopts "regular=T3+T1" and "normal=T4+T1" .

The standpoint of [a1] seems to be the most widely accepted.

The adjective "completely regular" is often associated with the letter $T_{3\frac12}$.

The axiom $T_D$ is intermediate between $T_0$ and $T_1$: it states that every singleton is a locally closed set.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)
[a2]	J.L. Kelley, "General topology" , Springer (1975)
[a3]	B. von Querenburg, "Mengentheoretische Topologie" , Springer (1973)
[b1]	Peter T. Johnstone Sketches of an elephant Oxford University Press (2002) ISBN 0198534256 Zbl 1071.18001