nonHausdorff

Non-Hausdorff spaces are topological spaces that do not satisfy the Hausdorff separation axiom. In a Hausdorff space, any two distinct points x and y can be separated by disjoint open neighborhoods. A non-Hausdorff space lacks this property for at least one pair of distinct points, and as a result the usual intuition about separation and convergence can fail.

This failure often leads to practical consequences in topology. For example, limits of nets or filters may not be unique when they exist, and the specialization preorder (where one point lies in the closure of another) becomes meaningful in ways that do not occur in Hausdorff spaces. Non-Hausdorff spaces frequently arise in quotient constructions, certain order topologies, and in spaces where points cannot be cleanly separated by neighborhoods.

Common examples illustrate the concept. The indiscrete topology on any set is non-Hausdorff because the only nonempty open set is the whole space, so distinct points cannot be separated. The particular-point topology selects a distinguished point p and declares every open set to contain p; this makes it impossible to separate p from some other point. The Sierpinski space on two points, and the cofinite topology on an infinite set, are also non-Hausdorff. The line with two origins is a classic manifold-like example that is locally Euclidean but globally non-Hausdorff. In algebraic geometry, spaces equipped with the Zariski topology, such as Spec of a ring, are generally not Hausdorff, though they are often T0 or T1.

Non-Hausdorff phenomena are often studied to understand how much topology relies on the separation axioms, and how to recover Hausdorff-like behavior through quotients or refinements when needed.