Hausdorff Space - Equivalences

Equivalences

For a topological space X, the following are equivalent:

  • X is a Hausdorff space.
  • Limits of nets in X are unique.
  • Limits of filters on X are unique.
  • Any singleton set {x} ⊂ X is equal to the intersection of all closed neighbourhoods of x. (A closed neighbourhood of x is a closed set that contains an open set containing x.)
  • The diagonal Δ = {(x,x) | xX} is closed as a subset of the product space X × X.

Read more about this topic:  Hausdorff Space

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