Hausdorff space

A Hausdorff space is one in which any two distinct points can be quarantined: each gets its own open bubble, and the two bubbles never touch. This is the modest separation hypothesis that makes a topological space behave like the spaces from analysis, where distinct points are obviously far enough apart to be told cleanly apart.

Formally, X is Hausdorff (also called T2) if for any two distinct points p and q there exist disjoint open sets U and V with p in U and q in V. Equivalently, distinct points have disjoint neighborhoods. The condition is mild but consequential: it is what is needed to make limits well-behaved.

In a Hausdorff space limits of sequences (and of nets) are unique — a convergent sequence cannot sneak up on two different points at once — and every compact subset is closed and every finite set is closed. Without the Hausdorff axiom strange things happen: in the indiscrete topology every sequence converges to every point, so limits are hopelessly ambiguous. Every metric space is Hausdorff (take balls of radius half the distance), which is why analysts rarely meet a non-Hausdorff space, but the axiom is genuinely an extra assumption, not a theorem.