Characterizations of Ideal Cluster Points

TL;DR

This paper explores the properties of ideal cluster points in topological spaces, establishing characterizations and relationships with classical cluster points and the topology generated by ideals.

Contribution

It provides new characterizations of ideal cluster points and links them to classical cluster points and the topology generated by ideals.

Findings

Characterization of $ ext{I}$-convergent sequences via subsequences.

Representation of $ ext{I}$-cluster points as classical cluster points.

Topology $ au$ coincides with the topology generated by $( au, ext{I})$.

Abstract

Given an ideal $\mathcal{I}$ on $\omega$, we prove that a sequence in a topological space $X$ is $\mathcal{I}$-convergent if and only if there exists a ``big'' $\mathcal{I}$-convergent subsequence. Then, we study several properties and show two characterizations of the set of $\mathcal{I}$-cluster points as classical cluster points of a filters on $X$ and as the smallest closed set containing ``almost all'' the sequence. As a consequence, we obtain that the underlying topology $\tau$ coincides with the topology generated by the pair $(\tau,\mathcal{I})$.