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hemicompact

Last-modified: 2010-12-04 (土) 14:57:06

Definition Edit

A topological space is said to be hemicompact if there is a sequence of compact subsets (called admissible sequence) such that every compact subset is contained in some member of the sequence.

Property Edit

  • A hemicompact space is the union of the admissible sequence since every singleton in the space is a compact set.
  • Every first countable hemicompact space is locally compact.
  • If X is hemicompact, then the space C(X) of all continuous functions from X to the real is metrizable. The metric is given by
    imgtex.fcgi?%5bres=100%5d%7b%5c%5bd_n%28f%2cg%29=%5csup_%7bx%5cin%20K_n%7d%7cf%28x%29-g%28x%29%7c%2c%5cquad%20d%28f%2cg%29=%5cfrac%7b1%7d%7b2%5en%7d%5cfrac%7bd_n%28f%2cg%29%7d%7b1+d_n%28f%2cg%29%7d%5c%5d%7d%25.png
    where K_n are the admissible sequence.

Reference Edit

Willard, Stephen, General Topology, Dover (2004).