On the Relationship between Ideal Cluster Points and Ideal Limit Points

Abstract

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 $F_\sigma$-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 $F_\sigma$-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 $F_\sigma$-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 $F_\sigma$- nor analytic P-ideal.