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countably ν-compact

Last-modified: 2010-12-02 (木) 22:06:47

Definition Edit

  • Topological space X is said to be countably ν-compact space if every countable ν-open cover of it has a finite sub cover.

Property Edit

  • ν-closed subset of a countably ν-compact space is countably ν-compact.
  • A ν-irresolute image of a countably ν-compact space is countably ν-compact.
  • countable product of countably ν-compact spaces is countably ν-compact.
  • countable union of countably ν-compact spaces is countably ν-compact.
  • For a ν-T_1 topological space X the following statements are equivalent
    1. X is countably ν-compact.
    2. Every countable family of ν-closed subsets of X which has the finite intersection property has a non-empty intersection.
    3. Every infinite subset has an ν-accumulation point.
    4. Every sequence in X has a ν-limit point.
    5. Every infinite ν-open cover has a proper sub cover

Reference Edit

  • S. Balasubramanian, P. Aruna Swathi Vyjayanthi and C. Sandhya,ν-Compact spaces , Scientia Magna, international book series, Vol. 5 (2009), No. 1 (78-82)