The Baire Category of Subsequences and Permutations which preserve Limit Points

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. \textbf{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.


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., for every sequence (A n ) of sets in I there exists A ∈ I such that A n \ 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 F σ , analytic, etc. Among the most important ideals, we find the family of asymptotic density zero sets for each neighborhood U of ℓ. The set of I-cluster points of x is denoted by Γ x (I). Moreover, ℓ ∈ X is an I-limit point of x if there exists a subsequence (x n k ) such that lim k→∞ x n k = ℓ and {n k : k ∈ N} / ∈ I.
The set of I-limit points is denoted by Λ x (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. [8,10,14,17,26,27]. It is worth noting that ideal cluster points have been studied much before under a different name. Indeed, as it follows by [23,Theorem 4.2], they correspond to classical "cluster points" of a filter F (depending on x) on the underlying space, cf. [6, Definition 2, p.69]. Lastly, let L x := Γ x (Fin) denote the set of ordinary limit points of x. Hence Λ x (I) ⊆ Γ x (I) ⊆ L x . See [23] for characterizations of I-cluster points and [2] for their relation with I-limit points. Let Σ be the sets of strictly increasing functions on N, that is, also, let Π be the sets of permutations of N, that is, Note that both Σ and Π are G δ -subsets of the Polish space N N , hence they are Polish spaces as well by Alexandrov's theorem; in particular, they are not meager in themselves, cf. [30,Chapter 2]. Given a sequence x and σ ∈ Σ, we denote by σ(x) the subsequence (x σ(n) ). Similarly, given π ∈ Π, we write π(x) for the rearranged sequence (x π(n) ). This gives clearly a bijection between Σ [resp. Π] and the set of subsequences of x [resp. permutations of x], cf. [1,3,27]. We will show that if I is a meager ideal and x is a sequence with values in a separable metric space then the set of subsequences (and permutations) 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 also an I-cluster point of x (Theorem 2.2). A similar result holds for I-limit points, provided that I is an analytic P-ideal (Theorem 2.6). Putting all together, this strenghtens all the results contained in [22] and answers an open question therein. As a byproduct, we obtain a characterization of meager ideals (Proposition 3.1). Lastly, the analogue statements fails for all maximal ideals (Example 2.3).

Main results
2.1. I-cluster points. It has been shown in [22] that, from a topological viewpoint, almost all subsequences of x preserve the set of I-cluster points, provided that I is "well separated" from its dual filter I ⋆ := {A ⊆ N : A c ∈ I}; that is, is comeager, cf. also [24] for the case I = Z and [20] for a measure theoretic analogue. We will extend this result to all meager ideals. In addition, we will see that the same holds also for Π x (I) := π ∈ Π : Γ π(x) (I) = Γ x (I) .
Here, given A, B, C ⊆ P(N), we say that A is separated from C by B if A ⊆ B and B ∩ C = ∅. In particular, an ideal I is F σ -separated from its dual filter I ⋆ if there exists an F σ -set B ⊆ P(N) such that I ⊆ B and B ∩ I ⋆ = ∅ (with the language of [9,18], the filter I ⋆ has rank ≤ 1). As it has been shown in [29, Corollary 1.5], the family of ideals I which are F σ -separated from I ⋆ includes all F σδ -ideals. In addition, a Borel ideal is F σseparated from its dual filter if and only if it does not contain an isomorphic copy of Fin × Fin (which can be represented as an ideal on N as where ν 2 (n) stands for the 2-adic valuation of n), see [19,Theorem 4]. In particular, Fin × Fin is a F σδσ -ideal which is not F σ -separated from its dual filter. For related results on F σ -separation, see [11,Proposition 3.6] and [32].
We show that the analogue of Theorem 2.1 holds for all meager ideals. In particular, this includes new cases as, for instance, I = Fin × Fin.
It is worth noting that every meager ideal I is F σ -separated from the Fréchet filter Fin ⋆ (see Proposition 3.1 below), hence I is F σ -separated from I ⋆ . This implies that our result is a proper generalization of Theorem 2.1: Theorem 2.2. Let x be a sequence in a first countable space X such that all closed sets are separable and let I be a meager ideal. Then the following are equivalent: Note that the standing hypotheses hold if X is a separable metric space. At this point, one may ask whether the same statement holds for all ideals. We show in the following example that the answer is negative. Example 2.3. Let I be a maximal ideal. Hence there exists a unique A ∈ {2N + 1, 2N + 2} such that A ∈ I. Set X = R. Let x be the characteristic function of A, i.e., x n = 1 if n ∈ A and x n = 0 otherwise. Then x → I 0. In A similar example can be found for Π x (I), replacing the embedding T with the homeomorphism H : Π → Π defined by H(π)(2n) = 2n−1 and H(π)(2n−1) = 2n for all n ∈ N.
As an application of our results, if x is I-convergent to ℓ, then the set of subsequences [resp., rearrangements] of x which are I-convergent to ℓ is not meager if and only if x is convergent (in the classical sense) to ℓ. (Here, a sequence x is said to be I-convergent to ℓ, shortened as . Let x be a sequence in a first countable compact space X. Let I be a meager ideal and assume that x is I-convergent to ℓ ∈ X. Then the following are equivalent: The proofs of Theorem 2.2 and Corollary 2.4 follow in Section 3.
2.2. I-limit points. Given a sequence x and an ideal I, definẽ and its analogue for permutations It has been shown in [22] that, in the case of I-limit points, the counterpart of Theorem 2.1 holds for generalized density ideals. Here, an ideal I is said to be a generalized density ideal if there exists a sequence (µ n ) of submeasures with finite and pairwise disjoint supports such that I = {A ⊆ N : lim n µ n (A) = 0}. More precisely: show that the answer is affirmative.
Note that this is strict generalization, as every generalized density ideal is an analytic P-ideal and there exists an analytic P-ideal which is not a generalized density ideal, see e.g. [5]. In addition, the same result holds for permutations.
Theorem 2.6. Let x be a sequence in a first countable space X such that all closed sets are separable and let I be an analytic P-ideal. Then the following are equivalent: Note that the same Example 2.3 shows that the analogue of Theorem 2.6 fails for all maximal ideals. The proof of Theorem 2.6 follows in Section 4.
We leave as an open question to check whether Theorem 2.6 may be extended to all meager ideals.

Proofs for I-cluster points
We start with a characterization of meager ideals (to the best of our knowledge, conditions (m3) and (m4) are novel). Here, a set A ⊆ P(N) is called hereditary if it is closed under subsets.
Note that each F k is closed and it does not contain any cofinite set. Therefore I is separated from Fin ⋆ by the F σ -set F . (m3) ⇐⇒ (m4) Suppose that I is separated from Fin ⋆ by k C k , where each C k is closed. Then it is enough to set In particular, F k does not contain any cofinite set. The converse is obvious.
(m3) =⇒ (m1) Suppose that there exists a sequence (F k ) of closed sets in {0, 1} N such that I ⊆ F := k F k and F ∩ Fin ⋆ = ∅. Then each F k has empty interior (otherwise it would contain a cofinite set). We conclude that I is contained in a countable union of nowhere dense sets.
The above characterization is reminescent of an open question of Mazur [25], cf. also [29, p. 220]: Is every F σδ -ideal contained in a hereditary F σ -set F such that X ∪ Y is not cofinite for all X, Y ∈ F ?
In addition, it is clear that condition (m3) is weaker than the extendability of I to a F σ -ideal. For characterizations and related results of the latter property, see e.g. [16,Theorem 4.4] and [12, Theorem 3.3].

Lemma 3.2. Let x be a sequence in a first countable space X and let I be a meager ideal. Then
Proof. Assume that L x = ∅, otherwise there is nothing to prove. Fix ℓ ∈ L x and let (U m ) be a decreasing local base at ℓ. Thanks to Proposition 3.1, there exists a sequence (F k ) of closed sets in {0, 1} N such that I is contained in k F k and F k ∩ Fin ⋆ = ∅ for all k ∈ N. At this point, we need to show that M := {σ ∈ Σ : ℓ / ∈ Γ σ(x) (I)} is meager. Observe that M ⊆ t k M t,k , where ∀t, k ∈ N, M t,k := σ ∈ Σ : n ∈ N : x σ(n) ∈ U t ∈ F k .
Hence, it is sufficient to show that each M t,k is nowhere dense.
Proof. The proof that P (ℓ) is comeager in Π is similar to the previous one, the only difference being in proving that the analogue of M t,k for permutations, that is, M t,k := {π ∈ Π : {n ∈ N : x π(n) ∈ U t } ∈ F k }, has empty interior, for each t, k ∈ N, where F k is defined as in (1). Let us suppose for the sake of contradiction that M t,k has an interior point, let us say π 0 . Then there exists n 0 ∈ N such that {π ∈ Π : π ↾ {1, . . . , n 0 } = π 0 ↾ {1, . . . , n 0 }} ⊆ M t,k .
We are finally ready to prove Theorem 2.2.

Remark 3.4.
As it is evident from the proof above, the hypothesis that "closed sets of X are separable" can be removed if, in addition, L x is countable.

1]. Note that
· ϕ is a submeasure which is invariant modulo finite sets. Moreover, replacing ϕ with ϕ/ N ϕ in (2), we can assume without loss of generality that N ϕ = 1.
Given a sequence x in a first countable topological space X and an analytic P-ideal I = Exh(ϕ), we define the function where (U k ) is a decreasing local base of neighborhoods at ℓ ∈ X. Clearly, the limit in (3) exists and it is independent of the choice of (U k ). is either comeager or empty for each ℓ ∈ X and q ∈ (0, 1).
Conversely, fix ℓ ∈ X, σ ∈ Σ, and q > 0 such that σ ∈ V (ℓ, q), hence A k ϕ > q for all k ∈ N, where A k := {n ∈ N : x σ(n) ∈ U k }. Let us define recursively a sequence (F k ) of finite subsets of N as it follows. Pick F 1 ⊆ A 1 such that ϕ(F 1 ) ≥ q (which is possibile since ϕ is a lscsm); then, for each integer k ≥ 2, let F k be a finite subset of A k such that min F k > max F k−1 and ϕ(F k ) ≥ q (which is possible since A k \ [1, max F k−1 ] ϕ = A k ϕ > q). Let (y n ) be the increasing enumeration of the set k F k , and define τ ∈ Σ such that τ (n) = y n for all n. It follows by construction that Therefore ℓ ∈ Λ σ(x) (I, q) ⊆ Λ σ(x) (I), which concludes the proof.  S(ℓ, q) is comeager for each ℓ ∈ L x and q ∈ (0, 1).
Therefore G ′ is a nonempty open subset of G which is disjoint from W k,s .