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 $x \neq y$ , 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.

Hausdorff spaces

T1

Hausdorff

Example

Context