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paracompact topological space∗ mathcam† 2013-03-21 13:15:42 A topological space X is said to be paracompact if every open cover of X has a locally finite open refinement. In more detail, if (Ui )i∈I is any family of open subsets of X such that ∪i∈I Ui = X , then there exists another family (Vi )i∈I of open sets such that ∪i∈I Vi = X Vi ⊂ Ui for all i ∈ I and any specific x ∈ X is in Vi for only finitely many i. Some properties: • Any metric or metrizable space is paracompact (A. H. Stone). • Given an open cover of a paracompact space X, there exists a (continuous) partition of unity on X subordinate to that cover. • A paracompact , Hausdorff space is regular. • A compact or pseudometric space is paracompact. ∗ hParacompactTopologicalSpacei created: h2013-03-21i by: hmathcami version: h31540i Privacy setting: h1i hDefinitioni h54-00i h55-00i † 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
