Abstract
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. 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.
Highlights
A classical result of Buck [7] states that, if x is real sequence, “almost every” subsequence of x has the same set of ordinary limit points of the original sequence x, in a measure sense
We identify each subsequence of of x with the function σ ∈ Σ defined by σ(n) = kn for all n ∈ N and, each rearranged sequence (xπ(n)) with the permutation π ∈ Π, cf. [1, 3, 29]
We will show that if I is a meager ideal and x is a sequence with values in a separable metric space the set of subsequences of x which preserve the set of I-cluster points of x is not meager if and only if every ordinary limit point of x is an I-cluster point of x (Theorem 2.2)
Summary
A classical result of Buck [7] states that, if x is real sequence, “almost every” subsequence of x has the same set of ordinary limit points of the original sequence x, in a measure sense. The aim of this note is to prove its topological analogue and non-analogue in the context of ideal convergence. Let I be an ideal on the positive integers N, that is, a family a subsets of N closed under subsets and finite unions. It is assumed that I contains the ideal Fin of finite sets and it is different from the power set P(N). I is a P-ideal if it is σ-directed modulo finite sets, i.e., 171 Page 2 of 14
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