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m-quasicompact

Last-modified: 2010-12-04 (土) 16:36:16

Definition Edit

  • A topological space X is m-quasicompact if and only if each open cover U of X by cozero sets, with the cardinality of U at most m, has a finite subcover. (m is infinite cardinal.)

Property Edit

  • Completely regular Hausdorff space is m-quasicompact if and only if it is m-pseudocompact.
  • Let X be a linearly ordered space, and let m be an infinite carinal. The following are then equivalent:
    1. X is m-bounded
    2. X is m-compact
    3. X is m-pseudocompact
    4. X is m-quasicompact
    5. For each gap u of X, |ω_{α(u)}|>m and |ω_{β(u)}|>m .

Reference Edit

Gulden, S. L. , Fleischman, W. M. and Weston, J. H. ,Linearly ordered topological spaces,Proc. Amer. Math. Soc. 24 1970 197--203.