Editing General topology (section)

==Separation axioms==
{{Main|Separation axiom}}
Many of these names have alternative meanings in some of mathematical literature, as explained on [[History of the separation axioms]]; for example, the meanings of "normal" and "T<sub>4</sub>" are sometimes interchanged, similarly "regular" and "T<sub>3</sub>", 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]].

* ''X'' is ''[[T0 space|T<sub>0</sub>]]'', or ''Kolmogorov'', if any two distinct points in ''X'' are [[topological distinguishability|topologically distinguishable]]. (It is a common theme among the separation axioms to have one version of an axiom that requires T<sub>0</sub> and one version that doesn't.)
* ''X'' is ''[[T1 space|T<sub>1</sub>]]'', or ''accessible'' or ''Fréchet'', if any two distinct points in ''X'' are separated. Thus, ''X'' is T<sub>1</sub> if and only if it is both T<sub>0</sub> and R<sub>0</sub>. (Though you may say such things as ''T<sub>1</sub> space'', ''Fréchet topology'', and ''Suppose that the topological space ''X'' is Fréchet'', avoid saying ''Fréchet space'' in this context, since there is another entirely different notion of [[Fréchet space]] in [[functional analysis]].)
* ''X'' is ''[[Hausdorff space|Hausdorff]]'', or ''T<sub>2</sub>'' or ''separated'', if any two distinct points in ''X'' are separated by neighbourhoods. Thus, ''X'' is Hausdorff if and only if it is both T<sub>0</sub> and R<sub>1</sub>. A Hausdorff space must also be T<sub>1</sub>.
* ''X'' is ''[[Urysohn and completely Hausdorff spaces|T<sub>2½</sub>]]'', or ''Urysohn'', if any two distinct points in ''X'' are separated by closed neighbourhoods. A T<sub>2½</sub> space must also be Hausdorff.
* ''X'' is ''[[regular space|regular]]'', or ''T<sub>3</sub>'', if it is T<sub>0</sub> and 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'' is also separated by closed neighbourhoods.)
* ''X'' is ''[[Tychonoff space|Tychonoff]]'', or ''T<sub>3½</sub>'', ''completely T<sub>3</sub>'', or ''completely regular'', if it is T<sub>0</sub> and if f, 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.
* ''X'' is ''[[normal space|normal]]'', or ''T<sub>4</sub>'', if it is Hausdorff and 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]].)
* ''X'' is ''[[completely normal space|completely normal]]'', or ''T<sub>5</sub>'' or ''completely T<sub>4</sub>'', if it is T<sub>1</sub> and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal.
* ''X'' is ''[[perfectly normal space|perfectly normal]]'', or ''T<sub>6</sub>'' or ''perfectly T<sub>4</sub>'', if it is T<sub>1</sub> and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff.

The [[Tietze extension theorem]]: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.