关于理想簇点与理想极限点的关系
On the Relationship between Ideal Cluster Points and Ideal Limit Points
摘要：
设$X$是第一可数空间，其中存在非平凡收敛序列，$\mathcal{I}$是解析P-理想.首先证明了$X$中所有序列的$\mathcal{I}$ -极限点的集合是闭的当且仅当$\mathcal{I}$也是$F_σ$ -理想.此外，设$(x_n)$是在没有孤立点的波兰空间中取值的序列。已知其统计极限点的集合$A$是$F_σ$ -集合，其统计聚点的集合$B$是闭的，其普通极限点的集合$C$是闭的，有$A\subseteq B\subseteq C$ .证明了集合$A$与$B$之间存在某种附加关系：$B$的孤立点集合$S$也包含在$A$中.相反，如果$A$是一个$F_σ$ -集，$B$是一个闭集，它有一个孤立点的子集$S$，使得$B\setminus S\neq \emptyset$是正则闭的，$C$是一个有$S\subseteq A\subseteq B\subseteq C$的闭集，那么存在一个序列$(x_n)$，其中：$A$是它的统计极限点的集合，$B$是它的统计聚类点的集合，$C$是它的普通极限点的集合。最后讨论了当$\mathcal{I}$既不是$F_σ$ -理想也不是解析P-理想时，$\mathcal{I}$ -极限点集的拓扑性质.
英文摘要：
Let $X$ be a first countable space which admits a non-trivial convergent sequence and let $\mathcal{I}$ be an analytic P-ideal. First, it is shown that the sets of $\mathcal{I}$-limit points of all sequences in $X$ are closed if and only if $\mathcal{I}$ is also an $F_σ$-ideal. Moreover, let $(x_n)$ be a sequence taking values in a Polish space without isolated points. It is known that the set $A$ of its statistical limit points is an $F_σ$-set, the set $B$ of its statistical cluster points is closed, and that the set $C$ of its ordinary limit points is closed, with $A\subseteq B\subseteq C$. It is proved the sets $A$ and $B$ own some additional relationship: indeed, the set $S$ of isolated points of $B$ is contained also in $A$. Conversely, if $A$ is an $F_σ$-set, $B$ is a closed set with a subset $S$ of isolated points such that $B\setminus S\neq \emptyset$ is regular closed, and $C$ is a closed set with $S\subseteq A\subseteq B\subseteq C$, then there exists a sequence $(x_n)$ for which: $A$ is the set of its statistical limit points, $B$ is the set of its statistical cluster points, and $C$ is the set of its ordinary limit points. Lastly, we discuss topological nature of the set of $\mathcal{I}$-limit points when $\mathcal{I}$ is neither $F_σ$- nor analytic P-ideal.
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