**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*) |*x*∈*X*} is closed as a subset of the product space*X*×*X*.

Read more about this topic: Hausdorff Space

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