关于拓扑空间中的弱函数空间、网络character和弱收敛
On meager function spaces, network character and meager convergence in topological spaces
摘要：
对于拓扑空间$X$的非孤立点$x$，网络特征$nw_χ(x)$是$X$的无限子集族的最小基数，使得$x$的每个邻域$O(x)$包含来自该族的集合。证明了：（1）每个无限紧Hausdorff空间$X$包含一个非孤立点$x$，该点具有$nw_χ(x)=\aleph_0$：（2）对于每个具有可数特征标的点$x\in X$，$X$中存在一个内射序列，对于$ω$上的某个稀疏滤子$\F$，该序列$\F$ -收敛到$x$;（3）若泛函Hausdorff空间$X$对于某个稀疏滤子$\F$包含一个$\F$ -收敛内射序列，则对于每个包含两个具有不相交闭包的非空开集的$T_1$ -空间$Y$，函数空间$C_p(X,Y)$是贫乏的。研究了在$βω$中允许单射$\F$ -收敛序列的滤子$\F$的性质.
英文摘要：
For a non-isolated point $x$ of a topological space $X$ the network character $nw_χ(x)$ is the smallest cardinality of a family of infinite subsets of $X$ such that each neighborhood $O(x)$ of $x$ contains a set from the family. We prove that (1) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $nw_χ(x)=\aleph_0$; (2) for each point $x\in X$ with countable character there is an injective sequence in $X$ that $\F$-converges to $x$ for some meager filter $\F$ on $ω$; (3) if a functionally Hausdorff space $X$ contains an $\F$-convergent injective sequence for some meager filter $\F$, then for every $T_1$-space $Y$ that contains two non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is meager. Also we investigate properties of filters $\F$ admitting an injective $\F$-convergent sequence in $βω$.
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