View source for Separation axiom

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A condition imposed on a [[Topological space|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 space]]; [[Kolmogorov axiom|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|Hausdorff space]]), and a $T_3$-space is called regular (cf. [[Regular space|Regular space]]); a Hausdorff $T_4$-space is always regular and is called normal (cf. [[Normal space|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|Completely regular space]]). A completely regular $T_2$-space is called a [[Tikhonov space]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR>
</table>



====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., [[#References|[a1]]].

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

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

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

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  B. von Querenburg,   "Mengentheoretische Topologie" , Springer  (1973)</TD></TR>
</table>

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

====References====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Peter T. Johnstone ''Sketches of an elephant'' Oxford University Press (2002)  ISBN 0198534256 {{ZBL|1071.18001}}</TD></TR>
</table>

[[Category:General topology]]