Limit points of subsequences

TL;DR

This paper investigates the conditions under which the collection of subsequences preserving certain limit points in a sequence is large, showing it is comeager precisely when these points match the ordinary limit points.

Contribution

It characterizes when the set of subsequences preserving $ ext{I}$-cluster or limit points is comeager, linking this to the equality with ordinary limit points.

Findings

The collection of subsequences preserving $ ext{I}$-cluster points is comeager if and only if they coincide with ordinary limit points.

Similarly, for $ ext{I}$-limit points, the collection is comeager under the same condition.

The collection of subsequences preserving ordinary limit points is always comeager.

Abstract

Let $x$ be a sequence taking values in a separable metric space and $\mathcal{I}$ be a generalized density ideal or an $F_\sigma$-ideal on the positive integers (in particular, $\mathcal{I}$ can be any Erd{\H o}s--Ulam ideal or any summable ideal). It is shown that the collection of subsequences of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$ [respectively, $\mathcal{I}$-limit points] is of second category if and only if the set of $\mathcal{I}$-cluster points of $x$ [resp., $\mathcal{I}$-limit points] coincides with the set of ordinary limit points of $x$; moreover, in this case, it is comeager. In particular, it follows that the collection of subsequences of $x$ which preserve the set of ordinary limit points of $x$ is comeager.