The Baire Category of Subsequences and Permutations which preserve Limit Points
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Results in Mathematics
The Baire Category of Subsequences and Permutations which preserve Limit Points Marek Balcerzak and Paolo Leonetti Abstract. 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 , then the set of subsequences [resp. permutations] which are I-convergent to is topologically large if and only if x is convergent to in the ordinary sense. Analogous results hold for I-limit points, provided I is an analytic P-ideal. Mathematics Subject Classification. Primary: 40A35, Secondary: 11B05, 54A20. Keywords. Ideal cluster points, Ideal limit points, Meager set, Analytic P-ideal, Ideal convergence, Subsequences, Permutations.
1. Introduction A classical result of Buck [7] states that, if x is real sequence, then “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. Unless otherwise stated, it is also 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., 0123456789().: V,-vol
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for every sequence (An ) of sets in I there exists A ∈ I such that An \ A is finite for all n. We regard ideals as subsets of the Cantor space {0, 1}N , hence we may speak about their topological complexities. In particular, an ideal can be Fσ , analytic, etc. Among the most important ideals, we find the family of asymptotic density zero sets Z := {A ⊆ N : limn→∞ n1 |A ∩ [1, n]| = 0}. We refer to [15] for a recent survey on ideals and associated filters. Let x = (xn ) be a sequence taking values in a topological space X, which will be always assumed to be Hausdorff. Then ∈ X is an I-cluster point of x if {n ∈ N : xn ∈ U } ∈ /I for each neighborhood U of . The set of I-cluster points of x is denoted by Γx (I). Moreover, ∈ X is an I-limit point of x if there exists a subsequence (xnk ) such that lim xnk = and {nk : k ∈ N} ∈ / I.
k→∞
The set of I-limit points is denoted by Λx (I). Statistical cluster points and statistical limits points (that is, Z-cluster points and Z-limit points) of real sequences were introduced by Fridy in [13] and studied by many authors, see e.g. [8,10,14,19,28,29]. It is worth noting that ideal cluster points have been studied much before under a different name. Indee
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