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\aleph-preparacompact

Last-modified: 2010-12-19 (日) 19:32:42

Name Edit

  • ℵ-preparacompact

Definition Edit

  • A Hausdorff space X is called preparacompact if each open cover of X has an open refinement imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cmathcal%7bH%7d=%5c%7bH_%7b%5calpha%7d%20:%20%5calpha%5cin%20A%5c%7d%20%5c%5d%7d%25.png such that if imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20B%5csubset%20A%20%5c%5d%7d%25.png is uncountable and if imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20p_%7b%5cbeta%7d%5cin%20H_%7b%5cbeta%7d%20%5c%5d%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20q_%7b%5cbeta%7d%5cin%20H_%7b%5cbeta%7d%20%5c%5d%7d%25.png for each imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cbeta%5cin%20B%20%5c%5d%7d%25.png with imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20p_%7b%5calpha%7d%5cneq%20p_%7b%5cbeta%7d%20%5c%5d%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20q_%7b%5calpha%7d%5cneq%20q_%7b%5cbeta%7d%20%5c%5d%7d%25.png for imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5calpha%5cneq%5cbeta%20%5c%5d%7d%25.png , then the set imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cmathcal%7bQ%7d=%5c%7bq_%7b%5cbeta%7d:%20%5cbeta%5cin%20B%5c%7d%20%5c%5d%7d%25.png has a limit point iff the set imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cmathcal%7bP%7d=%5c%7bp_%7b%5cbeta%7d:%20%5cbeta%5cin%20B%5c%7d%20%5c%5d%7d%25.png has a limit point.

Reference Edit

  • Davis, S.W. and Smith, J.C., The paracompactness of preparacompact spaces., Topology, Proc. Conf., Vol. 4, No.2, Ohio Univ. 1979, 345-360 (1980).