m-metacompact

Definition

m denotes an infinite cardinal.
The space X is said to be m-metacompact if and only if each open covering of X by no more than m sets admits as a refinement a point-finite open covering.

Property

• Let X be a linearly ordered space, and let m be an infinite cardinal. The following are then equivalent:
  1. X is m-fully normal.
  2. To each open covering U of X there corresponds a star-finite open covering V which is an m-quasi-refinement of U.
  3. X is almost m-fully normal.
  4. X is strongly m-paracompact.
  5. X is m-paracompact.
  6. X is m-metacompact.
  7. Each open covering U of X with |U|≦m admits as a refinement an open covering V which is point countable (that is, no point of X belongs to more than contably many members of V).
  8. For each open covering U of X with \aleph_0≦|U|≦m there is an open covering V which refines U such that each point of X belongs to less than |U| members of V.
  9. Each gap u of X satisfying |ω_α(u)|≦m (respectively,|ω_β(u)|≦m) is a Q-gap from the left (respectively, right).

Remark

Reference

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