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m-pseudocompact

Last-modified: 2010-12-04 (土) 14:17:56

Definition Edit

  • A topological space X is m-pseudocompact if each map (continuous function) f:X→R^m has a compact range, where R^m denotes the Cartesian product of the reals m times (m is a 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.