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relatively realcompact

Last-modified: 2010-10-25 (月) 01:41:24

Definition Edit

A subset A of a topological space X is called relatively realcompact in X iff imgtex.fcgi?%5bres=100%5d%7b$%5cmathrm%7bcl%7d_%7b%5cnu%20X%7dA%5csubset%20X$%7d%25.png, where νX denotes the Hewitt realcompactification of X.

Remark Edit

  • Recall that A is relatively compact? in X iff imgtex.fcgi?%5bres=100%5d%7b$%5cmathrm%7bcl%7d_X%20A$%7d%25.png is compact, and that A is relatively pseudocompact in X iff imgtex.fcgi?%5bres=100%5d%7b$%5cmathrm%7bcl%7d_%7b%5cbeta%20X%7dA%5csubset%20%5cnu%20X$%7d%25.png.
  • Note that if A is relatively realcompact in X, the closure of A in X is realcompact space. However, the converse is not true.

Reference Edit

  • J. J. Shommer, Relatively realcompact sets and nearly pseudocompact spaces, Comment. Math. Univ. Calorin. 34,2 (1993) 375-382.