Hausdorffness

Hausdorffness, or the Hausdorff property, is a separation condition in topology. A topological space X is Hausdorff (also called T2) if for any two distinct points x and y in X there exist open sets U and V with x ∈ U, y ∈ V and U ∩ V = ∅. Equivalently, distinct points can be separated by disjoint neighborhoods.

Consequences and basic properties include: in a Hausdorff space, singletons {x} are closed; in particular, finite sets are closed. A convergent sequence has at most one limit, and limits of nets are unique as well. The property is preserved by taking subspaces and products: subspaces of Hausdorff spaces are Hausdorff, and the product of Hausdorff spaces is Hausdorff. Any metric space is Hausdorff, so Euclidean spaces and more generally all normed spaces are Hausdorff.

Relation to other axioms and hierarchy: Hausdorffness implies the T1 separation axiom, but T1 does not imply Hausdorff. It sits within the standard separation hierarchy as T0, T1, T2, with additional properties such as regular and normal spaces refining these notions.

Examples and non-examples: Standard topologies on R^n are Hausdorff. The cofinite topology on an infinite set is not Hausdorff, nor is the indiscrete (trivial) topology.

Origin and naming: The term honors Felix Hausdorff, who introduced separation axioms and related concepts in the early 20th century.