Separation axioms

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 . 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 . 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

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Footnotes

1. German der hausdorffsche Raum ↩

2. 2010, Algebraische Topologie, p. 7 (Definition 1.1.25) ↩