Editing Hausdorff space (section)

== Properties ==
[[Subspace (topology)|Subspace]]s and [[product topology|products]] of Hausdorff spaces are Hausdorff, but [[quotient space (topology)|quotient space]]s of Hausdorff spaces need not be Hausdorff. In fact, ''every'' topological space can be realized as the quotient of some Hausdorff space.<ref>{{cite journal |last=Shimrat |first=M. |title=Decomposition spaces and separation properties |journal=[[Quarterly Journal of Mathematics]] |volume=2 |year=1956 |pages=128–129 |doi= 10.1093/qmath/7.1.128}}</ref>

Hausdorff spaces are [[T1 space|T<sub>1</sub>]], meaning that each [[singleton (mathematics)|singleton]] is a closed set. Similarly, preregular spaces are [[R0 space|R<sub>0</sub>]]. Every Hausdorff space is a [[Sober space]] although the converse is in general not true. 

Another property of Hausdorff spaces is that each [[compact set]] is a closed set. For non-Hausdorff spaces, it can be that each compact set is a closed set (for example, the [[cocountable topology]] on an uncountable set) or not (for example, the [[cofinite topology]] on an infinite set and the [[Sierpiński space]]). 

The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods,<ref>{{harvnb|Willard|2004|pp=124}}</ref> in other words there is a neighborhood of one set and a neighborhood of the other, such that the two neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.

Compactness conditions together with preregularity often imply stronger separation axioms. For example, any [[locally compact space|locally compact]] preregular space is [[completely regular space|completely regular]].{{sfn|Schechter|1996|loc=17.14(d), p. 460}}<ref>{{cite web |title=Locally compact preregular spaces are completely regular |url=https://math.stackexchange.com/questions/4503299 |website=math.stackexchange.com}}</ref> [[Compact space|Compact]] preregular spaces are [[normal space|normal]],{{sfn|Schechter|1996|loc=17.7(g), p. 457}} meaning that they satisfy [[Urysohn's lemma]] and the [[Tietze extension theorem]] and have [[partition of unity|partitions of unity]] subordinate to locally finite [[open cover]]s. The Hausdorff versions of these statements are: every locally compact Hausdorff space is [[Tychonoff space|Tychonoff]], and every compact Hausdorff space is normal Hausdorff.

The following results are some technical properties regarding maps ([[continuous (topology)|continuous]] and otherwise) to and from Hausdorff spaces.

Let ''<math>f\colon X \to Y</math>'' be a continuous function and suppose <math>Y</math> is Hausdorff. Then the [[Graph of a function|graph]] of ''<math>f</math>'', <math>\{(x,f(x)) \mid x\in X\}</math>, is a closed subset of ''<math>X \times Y</math>''.

Let ''<math>f\colon X \to Y</math>'' be a function and let <math>\ker(f) \triangleq \{(x,x') \mid f(x) = f(x')\}</math> be its [[kernel of a function|kernel]] regarded as a subspace of ''<math>X \times X</math>''.
*If ''<math>f</math>'' is continuous and ''<math>Y</math>'' is Hausdorff then ''<math>\ker(f)</math>'' is a closed set.
*If ''<math>f</math>'' is an [[open map|open]] surjection and ''<math>\ker(f)</math>'' is a closed set then ''<math>Y</math>'' is Hausdorff.
*If ''<math>f</math>'' is a continuous, open [[surjection]] (i.e. an open quotient map) then ''<math>Y</math>'' is Hausdorff [[if and only if]] ''<math>\ker(f)</math>'' is a closed set.

If ''<math>f, g \colon X \to Y</math>'' are continuous maps and ''<math>Y</math>'' is Hausdorff then the [[Equaliser (mathematics)|equalizer]] <math>\mbox{eq}(f,g) = \{x \mid f(x) = g(x)\}</math> is a closed set in ''<math>X</math>''. It follows that if ''<math>Y</math>'' is Hausdorff and ''<math>f</math>'' and ''<math>g</math>'' agree on a [[dense (topology)|dense]] subset of ''<math>X</math>'' then ''<math>f = g</math>''. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.

Let ''<math>f\colon X \to Y</math>'' be a [[closed map|closed]] surjection such that ''<math>f^{-1} (y)</math>'' is [[compact space|compact]] for all ''<math>y \in Y</math>''. Then if ''<math>X</math>'' is Hausdorff so is ''<math>Y</math>''.

Let ''<math>f\colon X \to Y</math>'' be a [[quotient map (topology)|quotient map]] with ''<math>X</math>'' a compact Hausdorff space. Then the following are equivalent:
*''<math>Y</math>'' is Hausdorff.
*''<math>f</math>'' is a [[closed map]].
*''<math>\ker(f)</math>'' is a closed set.