On the Relationship between Ideal Cluster Points and Ideal Limit Points

TL;DR

This paper explores the relationships between various types of limit points of sequences in topological spaces, characterizing when sets of these points are closed or regular, and examining the influence of ideal properties.

Contribution

It establishes conditions under which sets of $ ext{I}$-limit points are closed or regular, linking ideal properties with topological characteristics of limit point sets.

Findings

Sets of $ ext{I}$-limit points are closed iff $ ext{I}$ is an $F_\sigma$-ideal.

Characterization of statistical limit, cluster, and ordinary limit points in Polish spaces.

Existence of sequences with prescribed sets of limit and cluster points under certain conditions.

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.