| 摘要: |
设K为一类拓扑空间满足若X属于K则X的每个闭子空间也属于K，DK是可以表示为离散K闭集之并的拓扑空间类.我们证明了若空间X为次仿紧的K-散布空间且X中每个K闭集为Gδ-集，则X可表示为由K闭集构成的σ-离散集族之并.如果X是次仿紧K-散布空间，则X是DK-似空间，在上述结论中我们没有要求空间的分离性质，我们指出存在一个空间满足每个K闭集是Gδ-集但非所有DK闭集是Gδ-集.我们证明了若X是T1空间且是可数个ω-散布子空间的并，则X是半层空间当且仅当X是可数个闭离散子空间的并，这推广了文献[11]中定理3.9.应用上面的结论最后证明了如下结论：设H是半层Hausdorff空间，若H满足下面条件之一，则H是σ-空间：
a) H是散布或ω-散布空间；
b）H是可数个散布子空间的并；
c) H是可数个C-散布子空间的并；
d) H是可数个ω-散布子空间的并. |
| 关键词: K-散布空间 ω-散布空间 C-散布空间 次仿紧空间 K-似空间 |
| DOI： |
| 分类号:O189.1 |
| 基金项目:国家自然科学基金面上项目资助(12171015). |
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| A NOTE ON K-SCATTERED SPACES |
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PENG Liang-xue, HUANG Mu-wei, DENG Yu-ming
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School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China
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| Abstract: |
Let K be a class of topological spaces which are hereditary with respect to closed subspaces and let DK denote the class of all spaces which are discrete unions of spaces from K. We prove that if a space X is subparacompact K-scattered such that every closed set with property K is a Gδ-set of X, then X is a union of σ-discrete family of closed sets with property K. We show that if X is a subparacompact K-scattered space, then X is a DK-like space. The above result does not require any separation axioms for the space. We point out such that there is a space that every K-closed set is a Gδ-set, but not all DK-closed sets are Gδ-sets. We prove that if X is a T1-space and X is a union of countably many ω-scattered subspaces, then X is semi-stratifiable if and only if X is a union of countably many closed discrete subspaces, this result generalizes Theorem 3.9 in [11]. We finally get the following result: Suppose H is a semi-stratifiable Hausdorff space. Then H is a σ-space if any one of the following holds:
a) if H is scattered or ω-scattered;
b) if H is a union of countably many scattered subspaces of H;
c) if H is a union of countably many C-scattered subspaces of H;
d) if H is a union of countably many ω-scattered subspaces of H. |
| Key words: K-scattered space ω-scattered space C-scattered space subparacompact space K-like space |
