Category theoretical view of I-cluster and I-limit points of subsequences

Abstract

We study the concepts of I-limit and I-cluster points of a sequence, where I is an ideal with the Baire property. We obtain the relationship between I-limit and I-cluster points of a subsequence of a given sequence and the set of its classical limit points in the sense of category theory.

Some new insights into ideal convergence and subsequences

“…They have also been studied by Demirci [8] in detail. Note that some results on I-convergence, I-limit points and I-cluster points can be found in [2,8,14,18]. Definition 2.1.…”

“…Next we focus our attention on the relationship of sequences and their subsequences regarding their respective sets of I-cluster points, from the point of view of measure (category was treated, as already mentioned, in Theorem 2.3, [18]). These type of results have also been studied by Balcerzak and Leonetti [5].…”

“…It is well known that every x ∈ (0, 1] has a unique binary expansion x = ∑ ∞ n=1 2 −n d n (x) such that d n (x) = 1 for infinitely many positive integers n, and for every x ∈ (0, 1] and any sequence s = (s n ) we can generate a subsequence (sx) of s in such a way that: if d n (x) = 1, then (sx) n = s n . In the existing literature the relationships between a given sequence and its subsequences have been studied in two directions: the first direction is changing the concept of convergence by statistical convergence, A-statistical convergence, uniform statistical convergence, ideal convergence, and the other direction is using measure or category to study the measure and topological largeness of the sets of subsequences (see [2,4,15,[18][19][20][21][22][23][24]). There are still gaps to examine in this area.…”

Another characterization of meager ideals

“…By a known result due to Buck [7], almost every subsequence, in the sense of measure, of a given real sequence x has the same set of ordinary limit points of the original sequence x. Extensions and other measure-related results may be found in [1,12,13,16,17,18]. The aim of this note is to prove its topological [non]analogue in the context of ideal convergence, following the line of research in [3,14,15,19]. This will allow us to obtain a characterization of meager ideals in Theorem 1.…”

“…(i) [15,Theorem 2.3] for the case where X = R, I = Z := {A ⊆ N : lim n |A ∩ [1, n]|/n = 0}, and the equivalences (l1) ⇐⇒ (l2) ⇐⇒ (l5); (ii) [14, Theorem 2.3] for the case where I is a generalized density ideal and the equivalences (l1) ⇐⇒ (l2) ⇐⇒ (l5); (iii) [3, Theorem 2.9] for the case where I is an analytic P-ideal; (iv) [19,Theorem 1] for the case where X = R with the equivalence (l1) ⇐⇒ (l5). At this point, as it has been shown in [3, Example 2.6], it is worth noting that, if I is maximal (that is, the complement of a free ultrafilter), then there exists a bounded real sequence x which satisfies (c2) but not (c5), cf.…”