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σ-paracompact

Last-modified: 2010-07-22 (木) 14:46:01

Definition Edit

Suppose that if U is an open cover of a topological space X, there exests a sequence imgtex.fcgi?%5bres=100%5d%7b$V_n$%7d%25.png of open covers and for every point x in X, there exists an index m and W in U such that imgtex.fcgi?%5bres=100%5d%7b$%5cmathrm%7bSt%7d%28x%2cV_m%29%5csubset%20W$%7d%25.png.
Then X is called σ-compact.

Property Edit

  • Let X be a topological space. The following are equivalent.
    1. X is σ-paracompact.
    2. Every open cover of X has a σ-discrete closed refinement.
    3. Every open cover of X has a σ-locally-finite? closed refinement.
    4. Every open cover of X has a σ-closure-preserving? closed refinement.

Reference Edit

Dennis K. Burke, On subcompact spaces, Proc. Amer. Math. Soc. Vol.23 No.3 (1969) pp.655-663.