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semicompact

Last-modified: 2010-09-04 (土) 00:16:13

Definition 1 Edit

A topological space is called semicompact if every semiopen cover has a finite subcover.

Definition 2 Edit

Same as locally peripherally compact.

Property Edit

The following properties are all for Definition 1.

  • [Dorsett1981] A space is semicompact iff it satisfies the following.
    1. For every infinite subset S, imgtex.fcgi?%5bres=100%5d%7b$%5cmathrm%7bint%7d%28%5cmathrm%7bcl%7d%28S%29%29%5cneq%20%5cemptyset$%7d%25.png.
    2. Every disjoint family of nonempty open sets is finite.
  • [Ganster1987] A space X is semicompact iff it satisfies the following.
    1. X is S-closed.
    2. For every infinite subset S, imgtex.fcgi?%5bres=100%5d%7b$%5cmathrm%7bint%7d%28%5cmathrm%7bcl%7d%28S%29%29%5cneq%20%5cemptyset$%7d%25.png.

Reference Edit

Definition 1
  • Dorsett, Ch. Semi-compact R1 and product spaces, Bull. Malaysian Math. Soc., 3(2) (1980), 15-19.
  • Ch. Dorsett, Semi compactness, semi separation axioms, and product spaces, Bull. Malaysian Math. Soc. (2) 4 (1981), 21-28.
  • M. Ganster, Some remarks on strongly compact spaces and semi compact spaces, Bull. Malaysian Math. Soc. (10) 2 (1987), 67-81.
Definition 2
K. Morita, On closed mappings. II, Proc. Japan Acad. Vol.33, No.6 (1957) pp.325-327