Hausdorff space

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Definition

Given a Topological space [ilmath](X,\mathcal{J})[/ilmath] we say it is Hausdorff[1] 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][2]
(Unknown grade)
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.

References

  1. Introduction to Topology - Bert Mendelson
  2. Introduction to Topological Manifolds - John M. Lee