Hausdorff space

A Hausdorff space¹ or $T 2$ -space is a topological space $(𝑋, T)$ satisfying the separation axiom:²

For any $𝑥, 𝑦 \in 𝑋$ where $𝑥 \neq 𝑦$ , there exist open neighbourhoods $𝑈 \in T (𝑥)$ and $𝑉 \in T (𝑦)$ such that $𝑈 \cap 𝑉 = \emptyset$ . #m/def/topology

this can be easily generalised to a finite number of points:

For any finite set $𝐴 \subseteq 𝑋$ there exists an open neighbourhood $𝑈 𝑥$ of each $𝑥 \in 𝐴$ so that $𝑈 𝑥 \cap 𝑈 𝑦 = \emptyset$ for any $𝑥, 𝑦 \in 𝐴$ with $𝑥 \neq 𝑦$ . #m/thm/topology

Proof

Since $𝑋$ is hausdorff, for every $𝑥, 𝑦 \in 𝐴$ with $𝑥 \neq 𝑦$ there exists an open neighbourhood $𝑈 𝑥 𝑦$ of $𝑥$ and $𝑈 𝑦 𝑥$ of $𝑦$ so that $𝑈 𝑥 𝑦 \cap 𝑈 𝑦 𝑥 = \emptyset$ . For each $𝑥 \in 𝐴$ let $𝑈 𝑥 = ⋂ 𝑦 \in 𝐴 ∖ {𝑥} 𝑈 𝑥 𝑦$ . Then $𝑈 𝑥$ is an open neighbourhood of $𝑥$ and $𝑈 𝑥 \cap 𝑈 𝑦 𝑧 = \emptyset$ for every $𝑥, 𝑦, 𝑧 \in 𝐴$ with $𝑥 \neq 𝑦 \neq 𝑧$ . It follows that $𝑈 𝑥 \cap 𝑈 𝑦 = \emptyset$ for every $𝑥, 𝑦 \in 𝑋$ with $𝑥 \neq 𝑦$ .

Properties

A Hausdorff space guarantees uniqueness of the limit. If a space is first-countable, it is Hausdorff precisely when all limits are unique.
Hausdorffness is preserved by subspaces, products, and coproducts, but not quotients.
A space is Hausdorff iff the diagonal is closed

#state/develop | #lang/en | #SemBr

German der hausdorffsche Raum ↩
2010, Algebraische Topologie, p. 7 (Definition 1.1.25) ↩

Hausdorff space

Properties

Footnotes