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Separation axiom

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.

For the axiom of set theory, see Axiom schema of separation.

Separation axioms
in topological spaces

(Kolmogorov)

(Fréchet)

(Hausdorff)

(Urysohn)

(completely Hausdorff)

(regular Hausdorff)

(Tychonoff)

(normal Hausdorff)

(completely normal
 Hausdorff)

(perfectly normal
 Hausdorff)

The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties.


The precise definitions of the separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition.

X is , or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It will be a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.)

T0

X is , or symmetric, if any two topologically distinguishable points in X are separated.

R0

X is , or accessible or Fréchet, if any two distinct points in X are separated. Equivalently, every single-point set is a closed set. Thus, X is T1 if and only if it is both T0 and R0. (Although one may say such things as "T1 space", "Fréchet topology", and "suppose that the topological space X is Fréchet"; one should avoid saying "Fréchet space" in this context, since there is another entirely different notion of Fréchet space in functional analysis.)

T1

X is , or preregular, if any two topologically distinguishable points in X are separated by neighbourhoods. Every R1 space is also R0.

R1

X is , or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T0 and R1. Every Hausdorff space is also T1.

Hausdorff

X is , or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. Every T2½ space is also Hausdorff.

T2½

X is , or completely T2, if any two distinct points in X are separated by a continuous function. Every completely Hausdorff space is also T2½.

completely Hausdorff

X is if, given any point x and closed set F in X such that x does not belong to F, they are separated by neighbourhoods. (In fact, in a regular space, any such x and F will also be separated by closed neighbourhoods.) Every regular space is also R1.

regular

X is , or T3, if it is both T0 and regular.[1] Every regular Hausdorff space is also T2½.

regular Hausdorff

X is if, given any point x and closed set F in X such that x does not belong to F, they are separated by a continuous function.[2] Every completely regular space is also regular.

completely regular

X is , or T3½, completely T3, or completely regular Hausdorff, if it is both T0 and completely regular.[3] Every Tychonoff space is both regular Hausdorff and completely Hausdorff.

Tychonoff

X is if any two disjoint closed subsets of X are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.)

normal

X is if it is both R0 and normal. Every normal regular space is also completely regular.

normal regular

X is , or T4, if it is both T1 and normal. Every normal Hausdorff space is also both Tychonoff and normal regular.

normal Hausdorff

X is if any two separated sets are separated by neighbourhoods. Every completely normal space is also normal.

completely normal

X is , or T5 or completely T4, if it is both completely normal and T1. Every completely normal Hausdorff space is also normal Hausdorff.

completely normal Hausdorff

X is if any two disjoint closed sets are precisely separated by a continuous function. Every perfectly normal space is also both completely normal and completely regular.

perfectly normal

X is , or T6 or perfectly T4, if it is both perfectly normal and T0. Every perfectly normal Hausdorff space is also completely normal Hausdorff.

perfectly normal Hausdorff

These definitions all use essentially the preliminary definitions above.


Many of these names have alternative meanings in some of mathematical literature; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous.


Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.


In all of the following definitions, X is again a topological space.


The following table summarizes the separation axioms as well as the implications between them: cells which are merged represent equivalent properties, each axiom implies the ones in the cells to its left, and if we assume the T1 axiom, then each axiom also implies the ones in the cells above it (for example, all normal T1 spaces are also completely regular).

X is if, for every closed set C that is not the (possibly nondisjoint) union of two smaller closed sets, there is a unique point p such that the closure of {p} equals C. More briefly, every irreducible closed set has a unique generic point. Any Hausdorff space must be sober, and any sober space must be T0.

sober

X is if, for every continuous map f to X from a compact Hausdorff space, the image of f is closed in X. Any Hausdorff space must be weak Hausdorff, and any weak Hausdorff space must be T1.

weak Hausdorff

X is if the regular open sets form a base for the open sets of X. Any regular space must also be semiregular.

semiregular

X is if for any nonempty open set G, there is a nonempty open set H such that the closure of H is contained in G.

quasi-regular

X is if every open cover has an open star refinement. X is fully T4, or fully normal Hausdorff, if it is both T1 and fully normal. Every fully normal space is normal and every fully T4 space is T4. Moreover, one can show that every fully T4 space is paracompact. In fact, fully normal spaces actually have more to do with paracompactness than with the usual separation axioms.

fully normal

The axiom that all compact subsets are closed is strictly between T1 and T2 (Hausdorff) in strength. A space satisfying this axiom is necessarily T1 because every single-point set is necessarily compact and thus closed, but the reverse is not necessarily true; for the on infinitely many points, which is T1, every subset is compact but not every subset is closed. Furthermore, every T2 (Hausdorff) space satisfies the axiom that all compact subsets are closed, but the reverse is not necessarily true; for the cocountable topology on uncountably many points, the compact sets are all finite and hence all closed but the space is not T2 (Hausdorff).

cofinite topology

There are some other conditions on topological spaces that are sometimes classified with the separation axioms, but these don't fit in with the usual separation axioms as completely. Other than their definitions, they aren't discussed here; see their individual articles.

General topology

Schechter, Eric (1997). . San Diego: Academic Press. ISBN 0126227608. (has Ri axioms, among others)

Handbook of Analysis and its Foundations

Willard, Stephen (1970). . Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0-486-43479-6. (has all of the non-Ri axioms mentioned in the Main Definitions, with these definitions)

General topology

Separation Axioms at ProvenMath

from Schechter

Table of separation and metrisability axioms