\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
deepdream of
          a sidewalk
Show Question
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)
Math and science::Topology

Hausdorff spaces

Most interesting topological spaces are Hausdorff. Hausdorffness is identified as being the second separation condition. The first condition, T1 is a sub-requirement of T2 (Housdorff), so it is useful to keep it in mind when thinking about the Hausdorff condition.

T1

A topological space \( X \) is said to be \( T_1 \) iff every one-element subset of \( X \) is closed.

Now for Hausdorff.

Hausdorff

A topological space \( X \) is Hausdorff (or \( T_2 \)) iff for every distinct \( x, y \in X \), there exists disjoint neighbourhoods of \( x \) and \( y \).

Lemma. Every Hausdorff space is \( T_1 \).


Slightly more precise wording of the Hausdorff condition:

A topological space \( X \) is Hausdorff (or \( T_2 \)) iff for every \( x, y \in X \) such that \( x \ne y \), there exists disjoint open sets \( U, W \) of \( X \) such that \( x \in U \) and \( y \in W \).

Example

Every metrizable space is Hausdorff.

While most interesting spaces are Hausdorff, there are some non-Hausdorff spaces that are important. The Zariski topology is an example.

Context