The Baire Category of Subsequences and Permutations which preserve Limit Points

Abstract

AbstractLet $$\mathcal {I}$$ I be a meager ideal on $$\mathbf {N}$$ 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 $$\mathcal {I}$$ I -cluster points of x is topologically large if and only if every ordinary limit point of x is also an $$\mathcal {I}$$ 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 $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ , then the set of subsequences [resp. permutations] which are $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ is topologically large if and only if x is convergent to $$\ell $$ ℓ in the ordinary sense. Analogous results hold for $$\mathcal {I}$$ I -limit points, provided $$\mathcal {I}$$ I is an analytic P-ideal.