| 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，由此也可得若空间X是次仿紧K-散布空间且X中的每个DK闭集Gδ?-集，则X可表示为由K闭集构成的σ-离散集族之并。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。最后我们证明了若X是T_1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[9]中定理3.9。应用上面的结论可推广文献[2]中的定理1.4，我们把文献[2]定理1.4中H是散布空间的条件放宽到了H是可数个ω-散布(℃-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:国家自然科学基金项目（面上项目，重点项目，重大项目） |
|
| A note on K-scattered spaces |
|
Peng Liangxue
|
|
School of Mathematics, Statistics and Mechanics, Beijing University of Technology
|
| Abstract: |
| Let $\mathbb{K}$ be a class of topological spaces which are hereditary with respect
to closed subspaces and let $\mathbb{D}\mathbb{K}$ denote the class of all spaces which
are discrete unions of spaces from K. We prove that if a space $X$ is
subparacompact $\mathbb{K}$-scattered such that every closed set with property
$\mathbb{K}$ is a $G_{\delta}$-set of $X$, then $X$ is a union of $\sigma$-discrete
family of closed sets with property $\mathbb{K}$. We show that if $X$ is a
subparacompact $\mathbb{K}$-scattered space, then $X$ is a $\mathbb{DK}$-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 $\mathbb{K}$-closed set
is a $G_{\delta}$-set, but not all $\mathbb{D}\mathbb{K}$-closed sets are
$G_{\delta}$-sets. We prove that if $X$ is a $T_1$-space and $X$ is a
union of countably many $\omega$-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
\cite{NYIK}. We generalize Theorem 1.4 in \cite{MNC} by using the
conclusions above, the condition ``$H$ is scattered' in Theorem 1.4 in \cite{MNC} can
be weakened to $H$ is a union of countably many $\omega$-scattered ($\mathbb{C}$-scattered)
subspaces of $H$. We finally get the following result: Suppose $H$ is a semi-stratifiable
space with a monotonically normal compactification. Then $H$ is metrizable
if any one of the following holds:
\begin{enumerate}[ ]
\item a) if there is a $\sigma$-locally finite cover of $H$ by compact subsets；
\item b) if $H$ is scattered；
\item c) if $H$ is a union of countably many scattered subspaces of $H$；
\item d) if $H$ is a union of countably many $\mathbb{C}$-scattered subspaces of $H$；
\item e) if $H$ is a union of countably many $\omega$-scattered subspaces of $H$.
\end{enumerate} |
| Key words: K-scattered space ω-scattered space C-scattered space subparacompact space K-like space |
