Hausdorff space

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• Add Example:The real line with the finite complement topology is not Hausdorff as an example of a familiar set with an unfamiliar topology

Definition

Given a Topological space [ilmath](X,\mathcal{J})[/ilmath] we say it is Hausdorff or satisfies the Hausdorff axiom if:

• For all [ilmath]a,b\in X[/ilmath] that are distinct there exists neighbourhoods to [ilmath]a[/ilmath] and [ilmath]b[/ilmath], [ilmath]N_a[/ilmath] and [ilmath]N_b[/ilmath] such that:
  • [ilmath]N_a\cap N_b=\emptyset[/ilmath]

Alternate definition

• [ilmath]\forall a,b\in X\exists A,B\in\mathcal{J}[a\ne b\implies A\cap B=\emptyset][/ilmath]

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This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

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Are these statements the same? Clearly [ilmath]\text{neighbourhood }\implies\text{open-set} [/ilmath] as a neighbourhood to a point requires the existence of an open set containing that point (contained in the neighbourhood) and clearly [ilmath]\text{open-set}\implies\text{neighbourhood} [/ilmath] as an open set is a neighbourhood - write this up.

Further work for this page

• Link to a theorem about all metric spaces being Hausdorff.
• A subspace of a Hausdorff space is Hausdorff