Hausdorff space

A topological space T in which every two points u,v∈T have disjoint neighborhoods u∈U and v∈V separating them.

In a Hausdorff space, every point u∈T forms a closed set {u}.

Being Hausdorff is probably the single most important property a topological space can possess. So much so, it is often taken for granted.

Out of the separation axioms for topological spaces the Hausdorff condition is the most commonly encountered. It is weak enough that all "reasonable" spaces in analysis and differential geometry satisfy it, but strong enough to prevent pathological behavior. All metrizable spaces are Hausdorff (in fact, normal). Non-Hausdorff topologies are rare in most parts of pure mathematics. The simplest example of a non-Hausdorff topology is the indiscrete or trivial topology (on any space of more than one point), in which only the empty set and the entire space are open sets. A more interesting example is the Zariski topology of an algebraic variety which occurs in commutative algebra and algebraic geometry.

(Thanks Apatrix and Gorgonzola for edits.)