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m-metacompact

Last-modified: 2015-12-24 (ÌÚ) 04:37:17

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|¡åm 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¡å|U|¡åm 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 |¦Ø_¦Á(u)|¡åm (respectively,|¦Ø_¦Â(u)|¡åm) is a Q-gap from the left (respectively, right).

Reference

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