The Baire Category of Subsequences and Permutations which preserve Limit Points

Let $mathcal{I}$ be a meager ideal on $mathbf{N}$. We show that if $x$ is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of $x$ which preserve the set of $mathcal{I}$-cluster points of $x$ is topologically large if and only if every ordinary limit point of $x$ is also an $mathcal{I}$-cluster point of $x$. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. textbf{263} (2019), 221--229]. As an application, if $x$ is a sequence with values in a first countable compact space which is $mathcal{I}$-convergent to $ell$, then the set of subsequences [resp. permutations] which are $mathcal{I}$-convergent to $ell$ is topologically large if and only if $x$ is convergent to $ell$ in the ordinary sense. Analogous results hold for $mathcal{I}$-limit points, provided $mathcal{I}$ is an analytic P-ideal.