Invariance of Ideal Limit Points

TL;DR

This paper investigates the properties of ideal limit points of sequences in metric spaces, showing their set structure and invariance under subsequences for certain classes of ideals, including well-known density-based ideals.

Contribution

It establishes the topological nature of ideal limit points and proves their invariance under subsequences for a broad class of ideals satisfying specific conditions.

Findings

Set of ideal limit points is an $F_$-set or closed set.

Ideal limit points are invariant under subsequences for certain ideals.

Results apply to ideals with zero asymptotic or logarithmic density.

Abstract

Let $\mathcal{I}$ be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let $(x_n)$ be a sequence taking values in a metric space $X$. First, it is shown that the set of ideal limit points of $(x_n)$ is an $F_\sigma$-set [resp., a closet set]. Let us assume that $X$ is also separable and the ideal $\mathcal{I}$ satisfies certain additional assumptions, which however includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and some summable ideals. Then, it is shown that the set of ideal limit points of $(x_n)$ is equal to the set of ideal limit points of almost all its subsequences.