m-pseudocompact

Definition

• A topological space X is m-pseudocompact if each map (continuous function) f:X→R^m has a compact range, where R^m denotes the Cartesian product of the reals m times (m is a cardinal).

Property

• completely regular, Hausdorff space is m-quasicompact if and only if it is m-pseudocompact.
• Let X be a linearly ordered space, and let m be an infinite carinal. The following are then equivalent:
  1. X is m-bounded
  2. X is m-compact
  3. X is m-pseudocompact
  4. X is m-quasicompact
  5. For each gap u of X, |ω_{α(u)}|>m and |ω_{β(u)}|>m .

Reference

Gulden, S. L. , Fleischman, W. M. and Weston, J. H. ,Linearly ordered topological spaces,
Proc. Amer. Math. Soc. 24 1970 197--203.