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coparacompact

Last-modified: 2010-11-27 (土) 17:18:38

Definition Edit

A Hausdorff space X is called equi-locally convex (ELCX) if there exists an open neighborhood U of the diagonal imgtex.fcgi?%5bres=100%5d%7b$%5cDelta%5csubset%20X%5ctimes%20X$%7d%25.png, a mapping imgtex.fcgi?%5bres=100%5d%7b$%5cvarphi%20:U%5ctimes%20I%5cto%20X$%7d%25.png and an open cover imgtex.fcgi?%5bres=100%5d%7b$%5cmathcal%7bV%7d=%5c%7bV_%5calpha%5c%7d$%7d%25.png of X such that:

  1. for any t in I and x in X, imgtex.fcgi?%5bres=100%5d%7b$%5cvarphi%28x%2cx%2ct%29=x$%7d%25.png;
  2. for any x,y in X, imgtex.fcgi?%5bres=100%5d%7b$%5cvarphi%28x%2cy%2c0%29=x$%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b$%5cvarphi%28x%2cy%2c1%29=y$%7d%25.png;
  3. for any imgtex.fcgi?%5bres=100%5d%7b$V_%5calpha%20%5cin%20%5cmathcal%7bV%7d$%7d%25.png, imgtex.fcgi?%5bres=100%5d%7b$V_%5calpha%20%5ctimes%20V_%5calpha%20%5csubset%20U$%7d%25.png;
  4. for any imgtex.fcgi?%5bres=100%5d%7b$V_%5calpha%20%5cin%20%5cmathcal%7bV%7d$%7d%25.png, imgtex.fcgi?%5bres=100%5d%7b$%5cvarphi%28V_%5calpha%20%5ctimes%20V_%5calpha%20%5ctimes%20I%29%5csubset%20V_%5calpha$%7d%25.png.

A subset W in X is called convex if imgtex.fcgi?%5bres=100%5d%7b$W%5ctimes%20W%5csubset%20U$%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b$%5cvarphi%28W%5ctimes%20W%5ctimes%20I%29%5csubset%20W$%7d%25.png.
The cover imgtex.fcgi?%5bres=100%5d%7b$%5cmathcal%7bV%7d$%7d%25.png is called a convex open cover of X.

An ELCX space is called coparacompact if any open cover has a locally finite convex refinement.

Reference Edit

Vladimir P. Okhezin, On the fixed-point theory for non-compact maps and spaces.I, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center, Volume 5, 1995, 83-100.