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*Definition [#b865753d]
-A space X is said to be m-paracompact if each open covering of X by no more than m sets admits as a refinement a [[star-finite]] open covering.(m is infinite cardinal) [#n4108454]
*Property [#f697a120]
-Let X be a linearly ordered space, and let m be an infinite cardinal. The following are then equivalent: [#f85616ee]
++ X is m-fully normal.
++ To each open covering U of X there corresponds a star-finite open covering V which is an m-quasi-refinement of U.
++ X is almost m-fully normal.
++ X is strongly m-paracompact.
++ X is [[m-paracompact]].
++ X is [[m-metacompact]].
++ 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).
++ 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.
++ 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 [#l30acc6a]
-See [[strongly countably paracompact]]
*Reference [#b48f4aaf]
-Gulden, S. L. , Fleischman, W. M. and Weston, J. H., ''Linearly ordered topological spaces'', Proc. Amer. Math. Soc. 24 1970 197--203. [#re6f5d1f]
-M. K. Singal, ''Some Generalizations of Paracompactness'', Proceedings of the Kanpur topological conference, 1968. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 1971. pp. 245-263.
終了行:
*Definition [#b865753d]
-A space X is said to be m-paracompact if each open covering of X by no more than m sets admits as a refinement a [[star-finite]] open covering.(m is infinite cardinal) [#n4108454]
*Property [#f697a120]
-Let X be a linearly ordered space, and let m be an infinite cardinal. The following are then equivalent: [#f85616ee]
++ X is m-fully normal.
++ To each open covering U of X there corresponds a star-finite open covering V which is an m-quasi-refinement of U.
++ X is almost m-fully normal.
++ X is strongly m-paracompact.
++ X is [[m-paracompact]].
++ X is [[m-metacompact]].
++ 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).
++ 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.
++ 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 [#l30acc6a]
-See [[strongly countably paracompact]]
*Reference [#b48f4aaf]
-Gulden, S. L. , Fleischman, W. M. and Weston, J. H., ''Linearly ordered topological spaces'', Proc. Amer. Math. Soc. 24 1970 197--203. [#re6f5d1f]
-M. K. Singal, ''Some Generalizations of Paracompactness'', Proceedings of the Kanpur topological conference, 1968. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 1971. pp. 245-263.
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