T2Hausdorff

T2Hausdorff is a term sometimes used in topology to denote a space that satisfies the T2 separation axiom. In most works, T2 and Hausdorff are interchangeable, since a Hausdorff space is defined by the ability to separate any two distinct points with disjoint open neighborhoods. The expression T2Hausdorff is often used to emphasize this standard property or to clarify terminology in discussions of separation axioms.

Key properties and characterizations: A space X is T2Hausdorff if and only if for any distinct points x, y in X there exist disjoint open sets U and V with x in U and y in V. Equivalently, the diagonal Δ = { (x,x) ∈ X×X } is closed in the product X×X. This separation property is fundamental in many areas of topology and analysis, as it enables unique limits in many contexts and supports the construction of well-behaved function spaces.

Examples and non-examples: Any metric space is Hausdorff, so it is T2Hausdorff. The real numbers with the standard topology, finite discrete spaces, and any normed or more generally metric space are Hausdorff. Non-examples include spaces with the indiscrete (trivial) topology on an infinite set or the cofinite topology on an infinite set, both of which fail the separation requirement.

Relations to other axioms: T2Hausdorff is the same as the standard Hausdorff condition. It sits among the separation axioms as a stronger property than T1 or T0, and is weaker than some of the stronger separation conditions such as regularity or normality. See also separation axioms and Hausdorff spaces for broader context.