m-quasicompact

Definition

• A topological space X is m-quasicompact if and only if each open cover U of X by cozero sets, with the cardinality of U at most m, has a finite subcover. (m is infinite 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.