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