Editing Hausdorff space (section)

== Definitions ==
[[File:Hausdorff space.svg|thumb|203px|right|The points x and y, separated by their respective neighbourhoods U and V.]]

Points <math>x</math> and <math>y</math> in a topological space <math>X</math> can be ''[[separated by neighbourhoods]]'' if [[existential quantification|there exists]] a [[neighbourhood (topology)|neighbourhood]] <math>U</math> of <math>x</math> and a neighbourhood <math>V</math> of <math>y</math> such that <math>U</math> and <math>V</math> are [[Disjoint sets|disjoint]] <math>(U\cap V=\varnothing)</math>. <math>X</math> is a '''Hausdorff space''' if any two distinct points in <math>X</math> are separated by neighbourhoods. This condition is the third [[separation axiom]] (after T<sub>0</sub> and T<sub>1</sub>), which is why Hausdorff spaces are also called '''T<sub>2</sub> spaces'''. The name ''separated space'' is also used.

A related, but weaker, notion is that of a '''preregular space'''. <math>X</math> is a preregular space if any two [[topologically distinguishable]] points can be separated by disjoint neighbourhoods. A preregular space is also called an '''R<sub>1</sub> space'''.

The relationship between these two conditions is as follows. A topological space is Hausdorff [[if and only if]] it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and [[Kolmogorov space|Kolmogorov]] (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its [[Kolmogorov quotient]] is Hausdorff.