Compactness of certain bounded zero-sets in completely regular spaces
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- 作者：Mou ; Lei
- 关键词：54D30 ; 54D35 ; 54E18 ; Cech&ndash ; Stone compactification ; Dieudonné ; completion ; Zero-set ; Bounded ; Pseudocompact ; Regular Formula Not Shown -diagonal ; Metacompact ; M-space
- 刊名：Topology and its Applications
- 出版年：2005
- 出版时间：February 28, 2005
- 年：2005
- 卷：148
- 期：1-3
- 页码：153-163
- 全文大小：143 K
文摘
Let X be a completely regular Formula Not Shown -space. A zero-set Z in X is called a full zero-set if Formula Not Shown is a zero-set in the Cech–Stone compactification βX of X. As an generalization of theorems by W.G. McArthur and V. V. Uspenskii, we prove that every bounded, full zero-set F in X is compact if either (i) X has a regular Formula Not Shown -diagonal or (ii) X is a Baire space such that every open cover has a σ-point-finite open refinement. In case (i), F is metrizable by Šne?
der's theorem. We also apply this to show that if the Dieudonné completion μX of X is a paracompact M-space, then X is metrizable if either (i) or (iii) X is a Baire space with a σ-point-finite base.
der's theorem. We also apply this to show that if the Dieudonné completion μX of X is a paracompact M-space, then X is metrizable if either (i) or (iii) X is a Baire space with a σ-point-finite base.