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functionally compact

Last-modified: 2010-08-08 (日) 22:16:27

Definition Edit

  • A Hausdorff topological space X is called functionally compact if whenever U is an open filter base on X such that the intersection A of the elements of U is equal to the intersection of the closures of the elements of U, then U is a base for the neighborhoods of A.

Property Edit

  • A functionally compact Hausdorff space is compact if and only if it is regular.
  • A Hausdorff space X is functionally compact if and only if every mapping of X into any Hausdorff space is closed.
  • If X is C-compact then X is functionally compact.
  • Every functionally compact space is minimal Hausdorff.
  • Every CFC space is FFC and every FFC space is FC.

Remark Edit

Reference Edit

  • Dickman, R. F. and Jr. Zame, A.,Functionally compact spaces, Pacific J. Math. 31 1969 303--311
  • R.F. Dickman, Jr. and J.R. Porter, Between minimal Hausdorff and compact Hausdorff spaces, Topology Proc. Vol.9 (1984), pp.243-268.