Definition: A topological space X is said to be a Hausdorff space if and only if it satisfies the following Hausdorff Axiom:

• if x and y are distinct points of X then there exist open sets U and V such that x ∈ U, y ∈ V and U ∩ V = ∅.

Lemma 1.1 All metric spaces are Hausdorff spaces.

Proof Let X be a metric space with distance function d, and let x and y be points of X, where x ≠y. Let ε =1/2d(x, y). Then the open balls BX(x, ε)  and BX(y, ε) of radius ε centred on the points x and y are open sets (see Lemma 1.1). If BX(x, ε) ∩ BX(y, ε) were non-empty then there would exist z ∈ X satisfying d(x, z) < ε and d(z, y) < ε. But this is impossible, since it would then follow from the Triangle Inequality that d(x, y) < 2ε, contrary to the choice of ε. Thus x ∈ BX(x, ε), y ∈ BX(y, ε), BX(x, ε) ∩ BX(y, ε) = ∅.

This shows that the metric space X is a Hausdorff space.

We now give an example of a topological space which is not a Hausdorff space.