The Baire Category of Subsequences and Permutations which preserve Limit Points

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