Hausdorff space

A Hausdorff space¹ or -space is a topological space satisfying the separation axiom:²

For any where , there exist open neighbourhoods and such that . #m/def/topology

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

For any finite set there exists an open neighbourhood of each so that for any with . #m/thm/topology

Proof

Since is hausdorff, for every with there exists an open neighbourhood of and of so that . For each let . Then is an open neighbourhood of and for every with . It follows that for every with .

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