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Last-modified: 2010-12-30 (木) 18:44:42

Definition Edit

A topological space X is called paracompact if every open cover has a locally finite open refinement.

Property Edit

  • Every paracompact Hausdorff space is normal.
  • Every locally compact, paracompact Hausdorff space is strongly paracompact.
  • Every locally metrizable paracompact Hausdorff space is metrizable.
  • Every locally Cech-complete paracompact Haudorff space is a Cech-complete space.
  • Every Lindelof, countably paracompact space is paracompact.
  • Every metrizable space is paracompact.
  • More generally, Dowker characterized Hausdorff paracompact spaces as those spaces that have the property if U is an open cover of X then there exists a U-mapping f from X onto some metrizable space M. Here f is a U-mapping if f is continuous and there exists an open cover V of M such that f^{-1}(V) refines U. See D-paracompact, [Dowker1948] and [Nashef1991] for detail.
  • A Hausdorff space X is paracompact if and only if for each compactum Y the product X×Y is normal.
  • A regular T_1-space X is a paracompact if and only if each open cover has a locally finite.
  • Every collectionwise normal metacompact T_1-space is paracompact.
  • The following conditions are equivalent for a Hausdorff space X:
    1. the space X is paracompact;
    2. for each open cover of X there is a locally finite partition of unity subordinated to it;
    3. for each open cover of X there is a partition of unity subordinated to it.

Remark Edit

Reference Edit

  • 松島与三 多様体入門, 裳華房 (1965)
  • Engelking, General Topology, Taylor & Francis (June 1977)
  • C. H. Dowker, An extension of Alexandroff's mapping theorem, Bull. Am. Math. Soc. Vol.54, pp.386-391 (1948).
  • Bassam Al-Nashef, Cover-developements and D-paracompact spaces, Indian J. pure appl. Math. 22(2), pp.135-141, Feb. 1991.