Limit points of subsequences

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.Comment: To appear in Topology Appl. arXiv admin note: substantial text overlap with arXiv:1711.0426

Ideal version of Egorov's theorem for analytic P-ideals

AbstractWe introduce the notion of equi-ideal convergence and use it to prove an ideal variant of Egorov's theorem. We also show that this variant usually cannot be strengthen to a direct analogue of Egorov's theorem

Some remarks on II-faster convergent infinite series

summary:A structure of terms of $I$-faster convergent series is studied in the paper. Necessary and sufficient conditions for the existence of $I$-faster convergent series with different types of their terms are proved. Some consequences are discussed

Ideal Convergence of Sequences and Some of its Applications

We give a short survey of results on ideal convergence with some applications. In particular, we present a contribution of mathematicians from Łódź to these investigations during the recent 16 years