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Hausdorff space∗ yark† 2013-03-21 13:26:27 A topological space (X, τ ) is said to be T2 (or said to satisfy the T2 axiom) if given distinct x, y ∈ X, there exist disjoint open sets U, V ∈ τ (that is, U ∩ V = ∅) such that x ∈ U and y ∈ V . A T2 space is also known as a Hausdorff space. A Hausdorff topology for a set X is a topology τ such that (X, τ ) is a Hausdorff space. Properties The following properties are equivalent: 1. X is a Hausdorff space. 2. The set ∆ = {(x, y) ∈ X × X : x = y} is closed in the product topology of X × X. 3. For all x ∈ X, we have \ {x} = {A : A ⊆ X closed, ∃ open set U such that x ∈ U ⊆ A}. Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces. ∗ hHausdorffSpacei created: h2013-03-21i by: hyarki version: h31855i Privacy setting: h1i hDefinitioni h54D10i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1