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 Pideal.
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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 nonanalogue in the context of ideal convergence.
Let \(\mathcal {I}\) be an ideal on the positive integers \(\mathbf {N}\), that is, a family a subsets of \(\mathbf {N}\) closed under subsets and finite unions. Unless otherwise stated, it is also assumed that \(\mathcal {I}\) contains the ideal \(\mathrm {Fin}\) of finite sets and it is different from the power set \(\mathcal {P}(\mathbf {N})\). \(\mathcal {I}\) is a Pideal if it is \(\sigma \)directed modulo finite sets, i.e., for every sequence \((A_n)\) of sets in \(\mathcal {I}\) there exists \(A \in \mathcal {I}\) such that \(A_n\setminus A\) is finite for all n. We regard ideals as subsets of the Cantor space \(\{0,1\}^{\mathbf {N}}\), hence we may speak about their topological complexities. In particular, an ideal can be \(F_\sigma \), analytic, etc. Among the most important ideals, we find the family of asymptotic density zero sets
We refer to [15] for a recent survey on ideals and associated filters.
Let \(x=(x_n)\) be a sequence taking values in a topological space X, which will be always assumed to be Hausdorff. Then \(\ell \in X\) is an \(\mathcal {I}\)cluster point of x if
for each neighborhood U of \(\ell \). The set of \(\mathcal {I}\)cluster points of x is denoted by \(\Gamma _x(\mathcal {I})\). Moreover, \(\ell \in X\) is an \(\mathcal {I}\)limit point of x if there exists a subsequence \((x_{n_k})\) such that
The set of \(\mathcal {I}\)limit points is denoted by \(\Lambda _x(\mathcal {I})\). Statistical cluster points and statistical limits points (that is, \(\mathcal {Z}\)cluster points and \(\mathcal {Z}\)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. Indeed, as it follows by [26, Theorem 4.2], they correspond to classical “cluster points” of a filter \(\mathscr {F}\) (depending on x) on the underlying space, cf. [6, Definition 2, p.69]. Let also \(\mathrm {L}_x:=\Gamma _x(\mathrm {Fin})\) be the set of accumulation points of x, and note that \(\mathrm {L}_x=\Lambda _x(\mathrm {Fin})\) if X is first countable. Hence \(\Lambda _x(\mathcal {I})\subseteq \Gamma _x(\mathcal {I}) \subseteq \mathrm {L}_x\). We refer the reader to [26] for characterizations of \(\mathcal {I}\)cluster points and [2] for their relation with \(\mathcal {I}\)limit points. Lastly, we recall that the sequence x is said to be \(\mathcal {I}\)convergent to \(\ell \in X\), shortened as \(x \rightarrow _{\mathcal {I}} \ell \), if
for each neighborhood U of \(\ell \). Assuming that X is first countable, it follows by [26, Corollary 3.2] that, if \(x \rightarrow _{\mathcal {I}} \ell \) then \(\Lambda _x(\mathcal {I})=\Gamma _x(\mathcal {I})=\{\ell \}\), provided that \(\mathcal {I}\) is a Pideal. In addition, if X is also compact, then \(x \rightarrow _{\mathcal {I}} \ell \) is in fact equivalent to \(\Gamma _x(\mathcal {I})=\{\ell \}\), even if \(\mathcal {I}\) is not a Pideal, cf. [26, Corollary 3.4].
Let \(\Sigma \) be the sets of strictly increasing functions on \(\mathbf {N}\), that is,
also, let \(\Pi \) be the sets of permutations of \(\mathbf {N}\), that is,
Note that both \(\Sigma \) and \(\Pi \) are \(G_\delta \)subsets of the Polish space \(\mathbf {N}^{\mathbf {N}}\), hence they are Polish spaces as well by Alexandrov’s theorem; in particular, they are not meager in themselves, cf. [35, Chapter 2]. Given a sequence x and \(\sigma \in \Sigma \), we denote by \(\sigma (x)\) the subsequence \((x_{\sigma (n)})\). Similarly, given \(\pi \in \Pi \), we write \(\pi (x)\) for the rearranged sequence \((x_{\pi (n)})\). We identify each subsequence of \((x_{k_n})\) of x with the function \(\sigma \in \Sigma \) defined by \(\sigma (n)=k_n\) for all \(n \in \mathbf {N}\) and, similarly, each rearranged sequence \((x_{\pi (n)})\) with the permutation \(\pi \in \Pi \), cf. [1, 3, 29].
We will show that if \(\mathcal {I}\) 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 \(\mathcal {I}\)cluster points of x is not meager if and only if every ordinary limit point of x is also an \(\mathcal {I}\)cluster point of x (Theorem 2.2). A similar result holds for \(\mathcal {I}\)limit points, provided that \(\mathcal {I}\) is an analytic Pideal (Theorem 2.9). Putting all together, this strenghtens all the results contained in [25] 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 2.6).
2 Main Results
2.1 \(\mathcal {I}\)Cluster Points.
It has been shown in [25] that, from a topological viewpoint, almost all subsequences of x preserve the set of \(\mathcal {I}\)cluster points, provided that \(\mathcal {I}\) is “well separated” from its dual filter \(\mathcal {I}^\star :=\{A\subseteq \mathbf {N}: A^c \in \mathcal {I}\}\); that is,
is comeager, cf. also [27] for the case \(\mathcal {I}=\mathcal {Z}\) and [22] 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
Here, given \(\mathcal {A}, \mathcal {B}, \mathcal {C} \subseteq \mathcal {P}(\mathbf {N})\), we say that \(\mathcal {A}\) is separated from \(\mathcal {C}\) by \(\mathcal {B}\) if \(\mathcal {A}\subseteq \mathcal {B}\) and \(\mathcal {B} \cap \mathcal {C}=\emptyset \). In particular, an ideal \(\mathcal {I}\) is \(F_\sigma \)separated from its dual filter \(\mathcal {I}^\star \) if there exists an \(F_\sigma \)set \(\mathcal {B} \subseteq \mathcal {P}(\mathbf {N})\) such that \(\mathcal {I}\subseteq \mathcal {B}\) and \(\mathcal {B} \cap \mathcal {I}^\star =\emptyset \) (with the language of [9, 20], the filter \(\mathcal {I}^\star \) has rank \(\le 1\)).
Theorem 2.1
[25, Theorem 2.1] Let x be a sequence in a first countable space X such that all closed sets are separable and let \(\mathcal {I}\) be an ideal which is \(F_\sigma \)separated from its dual filter \(\mathcal {I}^\star \). Then \(\Sigma _x(\mathcal {I})\) is not meager if and only if \(\Gamma _x(\mathcal {I})=\mathrm {L}_x\). Moreover, in this case, it is comeager.
As it has been shown in [34, Corollary 1.5], the family of ideals \(\mathcal {I}\) which are \(F_{\sigma }\)separated from \(\mathcal {I}^\star \) includes all \(F_{\sigma \delta }\)ideals. In addition, a Borel ideal is \(F_\sigma \)separated from its dual filter if and only if it does not contain an isomorphic copy of \(\mathrm {Fin}\times \mathrm {Fin}\) (which can be represented as an ideal on \(\mathbf {N}\) as
where \(\nu _2(n)\) stands for the 2adic valuation of n), see [21, Theorem 4]. In particular, \(\mathrm {Fin}\times \mathrm {Fin}\) is a \(F_{\sigma \delta \sigma }\)ideal which is not \(F_\sigma \)separated from its dual filter. For related results on \(F_\sigma \)separation, see [11, Proposition 3.6] and [38].
We show that the analogue of Theorem 2.1 holds for all meager ideals. Hence, this includes new cases as, for instance, \(\mathcal {I}=\mathrm {Fin}\times \mathrm {Fin}\).
It is worth noting that every meager ideal \(\mathcal {I}\) is \(F_\sigma \)separated from the Fréchet filter \(\mathrm {Fin}^\star \) (see Proposition 3.1 below), hence \(\mathcal {I}\) is \(F_\sigma \)separated from \(\mathcal {I}^\star \). 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 \(\mathcal {I}\) be a meager ideal. Then the following are equivalent:
 (c1):

\(\Sigma _x(\mathcal {I})\) is comeager in \(\Sigma \);
 (c2):

\(\Sigma _x(\mathcal {I})\) is not meager in \(\Sigma \);
 (c3):

\(\Pi _x(\mathcal {I})\) is comeager in \(\Pi \);
 (c4):

\(\Pi _x(\mathcal {I})\) is not meager in \(\Pi \);
 (c5):

\(\Gamma _x(\mathcal {I})=\mathrm {L}_x\).
It is worth to spend some comments on the class \(\mathscr {C}\) of first countable spaces such that all closed sets are separable. It is clear that separable metric spaces belong to X. However, \(\mathscr {C}\) contains also nonmetrizable spaces:
Example 2.3
It has been shown by Ostaszewski in [31] that there exists a topological space X which is hereditarily separable (i.e., all subsets of X are separable), first countable, countably compact, and not compact, cf. also [17]. In particular, \(X \in \mathscr {C}\). However, X is not second countable (indeed, in the opposite, the notions of compactness and countably compactness would coincide). Considering that all separable metric spaces are second countable, it follows that X is not metrizable.
Moreover, there exists a separable first countable space outside \(\mathscr {C}\):
Example 2.4
Let X be the Sorgenfrey plane, that is, the product of two copies of the real line \(\mathbf {R}\) endowed with the lower limit topology. It is known that X is first countable and separable. Moreover, the antidiagonal \(\{(x,x): x \in \mathbf {R}\}\) is a closed uncountable discrete subspace of X, hence not separable, cf. [36, pp. 103–105]. Therefore \(X\notin \mathscr {C}\). A similar example is the Moore plane with the tangent disk topology, cf. [36, p. 176].
We can also show that there exists a first countable space outside \(\mathscr {C}\) which satisfies the statement of Theorem 2.2:
Example 2.5
Let X be an uncountable set, endowed the discrete topology. Then X is nonseparable first countable space, so that \(X\notin \mathscr {C}\). Thanks to Remark 3.4 below, Theorem 2.2 holds if the separability of all closed sets is replaced by the condition that \(\mathrm {L}_x\) is countable for each sequence x taking values in X. Indeed, the latter is verified because \(\mathrm {L}_x \subseteq \{x_n: n \in \mathbf {N}\}\).
We leave as open question whether there exists a topological space X (necessarily outside \(\mathscr {C}\)) and a meager ideal \(\mathcal {I}\) for which Theorem 2.2 fails. It is well possible that our main result extends beyond \(\mathscr {C}\), as it happened very recently with related results, see e.g. the improvement of [23, Theorem 4.2] in [18, Lemma 2.2].
Lastly, one may ask whether Theorem 2.2 holds for all ideals. We show in the following example that the answer is negative:
Example 2.6
Let \(\mathcal {I}\) be a maximal ideal. Hence there exists a unique \(A \in \{2\mathbf {N}+1,2\mathbf {N}+2\}\) such that \(A \in \mathcal {I}\). Set \(X=\mathbf {R}\). Let x be the characteristic function of A, i.e., \(x_n=1\) if \(n \in A\) and \(x_n=0\) otherwise. Then \(x \rightarrow _{\mathcal {I}} 0\). In particular, \(\Gamma _x(\mathcal {I})=\{0\}\). Note that a subsequence \(\sigma (x)\) is \(\mathcal {I}\)convergent to 0 if and only if \(\Gamma _{\sigma (x)}=\{0\}\). Then
Considering that \(\sigma ^{1}[A] \cup \sigma ^{1}[A1]\) is cofinite, we have either \(\sigma ^{1}[A] \in \mathcal {I}\) or \(\sigma ^{1}[A1]\in \mathcal {I}\). Let \(T: \Sigma \rightarrow \Sigma \) be the embedding defined by \(\sigma \mapsto \sigma +1\), so that \(\Sigma \) is homeomorphic to the open set \(T[\Sigma ]=\{\sigma \in \Sigma : \sigma (1)\ge 2\}\). Notice that
which implies that the open set \(T[\Sigma ]\) is contained in \(\Sigma _x(\mathcal {I}) \cup T[\Sigma _x(\mathcal {I})]\). Therefore both \(\Sigma _x(\mathcal {I})\) and \(T[\Sigma _x(\mathcal {I})]\) are not meager.
A similar example can be found for \(\Pi _x(\mathcal {I})\), replacing the embedding T with the homeomorphism \(H: \Pi \rightarrow \Pi \) defined by \(H(\pi )(2n)=2n1\) and \(H(\pi )(2n1)=2n\) for all \(n \in \mathbf {N}\).
As noted by the referee, the equivalences (c1) \(\Longleftrightarrow \) (c2) and (c3) \(\Longleftrightarrow \) (c4), and their analogues in the next results, can be viewed a consequence a general topological \(0\text {}1\) law stating that every tail set with the property of Baire is either meager or comeager, see [32, Theorem 21.4]. However, it seems rather difficult to show that the tail sets \(\Sigma _x(\mathcal {I})\) and \(\Pi _x(\mathcal {I})\) have the property of Baire, provided that \(\mathcal {I}\) is meager (note that this is surely false if \(\mathcal {I}\) is a maximal ideal, as it follows by Example 2.6).
As an application of our results, if x is \(\mathcal {I}\)convergent to \(\ell \), then the set of subsequences [resp., rearrangements] of x which are \(\mathcal {I}\)convergent to \(\ell \) is not meager if and only if x is convergent (in the classical sense) to \(\ell \). This is somehow related to [1, Theorem 2.1] and [3, Theorem 1.1]; cf. also [28, Theorem 3] for a measure theoretical nonanalogue.
Corollary 2.7
Let x be a sequence in a first countable compact space X. Let \(\mathcal {I}\) be a meager ideal and assume that x is \(\mathcal {I}\)convergent to \(\ell \in X\). Then the following are equivalent:
 (i1):

\(\{\sigma \in \Sigma : \sigma (x) \rightarrow _{\mathcal {I}} \ell \}\) is comeager in \(\Sigma \);
 (i2):

\(\{\sigma \in \Sigma : \sigma (x) \rightarrow _{\mathcal {I}} \ell \}\) is not meager in \(\Pi \);
 (i3):

\(\{\pi \in \Pi : \pi (x) \rightarrow _{\mathcal {I}} \ell \}\) is comeager in \(\Sigma \);
 (i4):

\(\{\pi \in \Pi : \pi (x) \rightarrow _{\mathcal {I}} \ell \}\) is not meager in \(\Pi \);
 (i5):

\(\lim _n x_n=\ell \).
The proofs of Theorem 2.2 and Corollary 2.7 follow in Sect. 3.
2.2 \(\mathcal {I}\)Limit Points
Given a sequence x and an ideal \(\mathcal {I}\), define
and its analogue for permutations
It has been shown in [25] that, in the case of \(\mathcal {I}\)limit points, the counterpart of Theorem 2.1 holds for generalized density ideals. Here, an ideal \(\mathcal {I}\) is said to be a generalized density ideal if there exists a sequence \((\mu _n)\) of submeasures with finite and pairwise disjoint supports such that \(\mathcal {I}=\{A\subseteq \mathbf {N}: \lim _n \mu _n(A)=0\}\). More precisely:
Theorem 2.8
[25, Theorem 2.3] Let x be a sequence in a first countable space X such that all closed sets are separable and let \(\mathcal {I}\) be generalized density ideal. Then \(\tilde{\Sigma }_x(\mathcal {I})\) is not meager if and only if \(\Lambda _x(\mathcal {I})=\mathrm {L}_x\). Moreover, in this case, it is comeager.
See [24] for a measure theoretic analogue. It has been left as open question to check, in particular, whether the same statement holds for analytic Pideals. We show that the answer is affirmative.
Note that this is strict generalization, as every generalized density ideal is an analytic Pideal and there exists an analytic Pideal which is not a generalized density ideal, see e.g. [5]. In addition, the same result holds for permutations.
Theorem 2.9
Let x be a sequence in a first countable space X such that all closed sets are separable and let \(\mathcal {I}\) be an analytic Pideal. Then the following are equivalent:
 (L1):

\(\tilde{\Sigma }_x(\mathcal {I})\) is comeager in \(\Sigma \);
 (L2):

\(\tilde{\Sigma }_x(\mathcal {I})\) is not meager in \(\Sigma \);
 (L3):

\(\tilde{\Pi }_x(\mathcal {I})\) is comeager in \(\Pi \);
 (L4):

\(\tilde{\Pi }_x(\mathcal {I})\) is not meager in \(\Pi \);
 (L5):

\(\Gamma _x(\mathcal {I})=\mathrm {L}_x\).
Note that the same Example 2.6 shows that the analogue of Theorem 2.9 fails for all maximal ideals. The proof of Theorem 2.9 follows in Sect. 4.
We leave as an open question to check whether Theorem 2.9 may be extended to all meager ideals.
3 Proofs for \(\mathcal {I}\)Cluster Points
We start with a characterization of meager ideals (to the best of our knowledge, condition (m3) is novel). Here, a set \(\mathcal {A}\subseteq \mathcal {P}(\mathbf {N})\) is called hereditary if it is closed under subsets.
Proposition 3.1
Let \(\mathcal {I}\) be an ideal on \(\mathbf {N}\). Then the following are equivalent:
 (m1):

\(\mathcal {I}\) is meager;
 (m2):

There exists a strictly increasing sequence \((\iota _n)\) of positive integers such that \(A \notin \mathcal {I}\) whenever \(\mathbf {N} \cap [\iota _n,\iota _{n+1}) \subseteq A\) for infinitely many \(n \in \mathbf {N}\);
 (m3):

\(\mathcal {I}\) is \(F_\sigma \)separated from the Fréchet filter \(\mathrm {Fin}^\star \).
Proof
(m1) \(\Longleftrightarrow \) (m2) See [37, Theorem 2.1]; cf. also [4, Theorem 4.1.2].
(m2) \(\implies \) (m3) Define \(I_n:=\mathbf {N} \cap [\iota _n,\iota _{n+1})\) for all \(n \in \mathbf {N}\). Then \(\mathcal {I}\subseteq F\), where \(F:=\bigcup _k F_k\) and
Note that each \(F_k\) is closed and it does not contain any cofinite set. Therefore \(\mathcal {I}\) is separated from \(\mathrm {Fin}^\star \) by the \(F_\sigma \)set F.
(m3) \(\implies \) (m1) Suppose that there exists a sequence \((F_k)\) of closed sets in \(\{0,1\}^{\mathbf {N}}\) such that \(\mathcal {I}\subseteq F:=\bigcup _k F_k\) and \(F \cap \mathrm {Fin}^\star =\emptyset \). Then each \(F_k\) has empty interior (otherwise it would contain a cofinite set). We conclude that \(\mathcal {I}\) is contained in a countable union of nowhere dense sets. \(\square \)
It is clear that condition (m3) is weaker than the extendability of \(\mathcal {I}\) to a \(F_{\sigma }\)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 \(\mathcal {I}\) be a meager ideal. Then
are comeager for each infinite set \(A\subseteq \mathbf {N}\).
Proof
Fix an infinite set \(A\subseteq \mathbf {N}\). As in the proof of Proposition 3.1, we can define intervals \(I_n:=\mathbf {N}\cap [\iota _n, \iota _{n+1})\) for all \(n \in \mathbf {N}\) such that a set \(S\subseteq \mathbf {N}\) does not belong to \(\mathcal {I}\) whenever S contains infinitely many intervals \(I_n\)s. At this point, for each \(n,k \in \mathbf {N}\), define the sets
Note that each \(X_k\) is open and that each \(Y_n\) is open and dense. Therefore the set \(\bigcap _n Y_n\), which can be rewritten as \(\{\sigma \in \Sigma : \sigma ^{1}[A]\notin \mathcal {I}\}\), is comeager. The proof that \(\{\pi \in \Pi : \pi ^{1}[A]\notin \mathcal {I}\}\) is comeager is analogous. \(\square \)
Lemma 3.3
Let x be a sequence in a first countable space X and let \(\mathcal {I}\) be a meager ideal. Then
are comeager for each \(\ell \in \mathrm {L}_x\).
Proof
Assume that \(\mathrm {L}_x\ne \emptyset \), otherwise there is nothing to prove. Fix \(\ell \in \mathrm {L}_x\) and let \((U_m)\) be a decreasing local base at \(\ell \). For each \(m \in \mathbf {N}\), define the infinite set \(A_m:=\{n \in \mathbf {N}: x_n \in U_m\}\). Thanks to Lemma 3.2, the set \(B_m:=\{\sigma \in \Sigma : \sigma ^{1}[A_m]\notin \mathcal {I}\}\) is comeager. Since \(S(\ell )\) can be rewritten as \(\bigcap _m B_m\), it follows that \(S(\ell )\) is comeager. The proof that \(P(\ell )\) is comeager is analogous.
\(\square \)
We are finally ready to prove Theorem 2.2.
Proof of Theorem 2.2
(c1) \(\implies \) (c2) It is obvious.
(c2) \(\implies \) (c5) Suppose that there exists \(\ell \in \mathrm {L}_x\setminus \Gamma _x(\mathcal {I})\). Then \(\Sigma _x(\mathcal {I})\) is contained in \(\Sigma \setminus S(\ell )\), which is meager by Lemma 3.3.
(c5) \(\implies \) (c1) Suppose that \(\mathrm {L}_x\ne \emptyset \), otherwise the claim is trivial. Let \(\mathscr {L}\) be a countable dense subset of \(\mathrm {L}_x\), so that \(\mathscr {L}\subseteq \Gamma _{\sigma (x)}(\mathcal {I})\) for each \(\sigma \in S:=\bigcap _{\ell \in \mathscr {L}}S(\ell )\), which is comeager by Lemma 3.3. Fix \(\sigma \in S\). On the one hand, \(\Gamma _{\sigma (x)}(\mathcal {I}) \subseteq \mathrm {L}_{\sigma (x)}\subseteq \mathrm {L}_x\). On the other hand, since \(\Gamma _{\sigma (x)}(\mathcal {I})\) is closed by [26, Lemma 3.1(iv)], we get \(\mathrm {L}_x\subseteq \Gamma _{\sigma (x)}(\mathcal {I})\). Therefore \(S\subseteq \Sigma _x(\mathcal {I})\).
The implications (c3) \(\implies \) (c4) \(\implies \) (c5) \(\implies \) (c3) are analogous. \(\square \)
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, \(\mathrm {L}_x\) is countable.
Lemma 3.5
Let \(\mathcal {I}\) be an ideal and x be a sequence in a first countable compact space. Then \(x \rightarrow _{\mathcal {I}} \ell \) if and only if \(\Gamma _x(\mathcal {I})=\{\ell \}\).
Proof
It follows by [26, Corollary 3.4]. \(\square \)
Proof of Corollary 2.7
(i1) \(\implies \) (i2) It is obvious.
(i2) \(\implies \) (i5) By Lemma 3.5, the hypothesis can be rewritten as \(\Gamma _x(\mathcal {I})=\{\ell \}\). Hence, condition (i2) is equivalent to the nonmeagerness of \(\{\sigma \in \Sigma : \Gamma _{\sigma (x)}=\Gamma _x(\mathcal {I})=\{\ell \}\}\). The claim follows by Theorem 2.2 and Remark 3.4.
(i5) \(\implies \) (i1) If \(x\rightarrow \ell \) then \(\sigma (x)\rightarrow _{\mathcal {I}} \ell \) for all \(\sigma \in \Sigma \).
The implications (i3) \(\implies \) (i4) \(\implies \) (i5) \(\implies \) (i3) are analogous. \(\square \)
4 Proofs for \(\mathcal {I}\)Limit Points
A lower semicontinuous submeasure (in short, lscsm) is a monotone subadditive function \(\varphi : \mathcal {P}(\mathbf {N}) \rightarrow [0,\infty ]\) such that \(\varphi (\emptyset )=0\), \(\varphi (F)<\infty \) for all \(F \in \mathrm {Fin}\), and \(\varphi (A)=\lim _n \varphi (A\cap [1,n])\) for all \(A\subseteq \mathbf {N}\). By a classical result of Solecki, an ideal \(\mathcal {I}\) is an analytic Pideal if and only if there exists a lscsm \(\varphi \) such that
where \(\Vert A\Vert _\varphi :=\lim _n \varphi (A\setminus [1,n])\) for all \(A\subseteq \mathbf {N}\), see [33, Theorem 3.1]. Note that \(\Vert \cdot \Vert _\varphi \) is a submeasure which is invariant modulo finite sets. Moreover, replacing \(\varphi \) with \(\varphi / \Vert \mathbf {N}\Vert _\varphi \) in (2), we can assume without loss of generality that \(\Vert \mathbf {N}\Vert _\varphi =1\).
Given a sequence x in a first countable topological space X and an analytic Pideal \(\mathcal {I}=\mathrm {Exh}(\varphi )\), we define the function
where \((U_k)\) is a decreasing local base of neighborhoods at \(\ell \in X\). Clearly, the limit in (3) exists and it is independent of the choice of \((U_k)\).
Lemma 4.1
Let x be a sequence in a first countable space X and let \(\mathcal {I}=\mathrm {Exh}(\varphi )\) be an analytic Pideal. Then, the section \(\mathfrak {u}(\sigma ,\cdot )\) is upper semicontinuous for each \(\sigma \in \Sigma \). In particular, the set
is closed for all \(q>0\).
Proof
See [2, Lemma 2.1]. \(\square \)
Lemma 4.2
With the same hypotheses of Lemma 4.1, the set
is either comeager or empty for each \(\ell \in X\) and \(q \in (0,1)\).
Proof
Suppose that \(V(\ell ,q) \ne \emptyset \), so that \(\ell \in \mathrm {L}_x\), and note that
Then, it is sufficient to show that
is nowhere dense for all \(k,s \in \mathbf {N}\).
To this aim, for every nonempty open set \(Z\subseteq \Sigma \), we need to prove that there exists a nonempty open subset \(S \subseteq Z\) such that \(S \cap W_{k,s}=\emptyset \). Fix a nonempty open set \(Z\subseteq \Sigma \) and \(\sigma _0 \in Z\) so that there exists \(n_0 \in \mathbf {N}\) for which
Since \(\ell \in \mathrm {L}_x\), there exists \(\sigma _1 \in Z^\prime \) such that \(\lim _n x_{\sigma _1(n)}=\ell \). Therefore
where \(n_1:=\max \{n_0+1,s\}\). At this point, since \(\varphi \) is a lscsm, it follows that there exists an integer \(n_2>n_1\) such that \(\varphi (\{n \in \mathbf {N} \cap [n_1, n_2]: x_{\sigma _1(n)} \in U_k\})> q\). Therefore \( S:=\{\sigma \in Z^\prime : \sigma \upharpoonright \{n_1,\ldots ,n_2\}=\sigma _1 \upharpoonright \{n_1,\ldots ,n_2\}\} \) is a nonempty open set contained in Z and disjoint from \(W_{k,s}\). Indeed
by the monotonicity of \(\varphi \). \(\square \)
Lemma 4.3
With the same hypotheses of Lemma 4.1, we have
In addition, \(\tilde{S}(\ell ,q):=\{\sigma \in \Sigma : \ell \in \Lambda _{\sigma (x)}(\mathcal {I},q)\}\) contains \(V(\ell ,q)\).
Proof
Fix \(\ell \in X\) and \(\sigma \in \tilde{S}(\ell )\), where
Then there exist \(\tau \in \Sigma \) and \(q>0\) such that \(\lim _n x_{\tau (\sigma (n))}=\ell \) and \(\Vert \tau (\mathbf {N})\Vert _\varphi \ge 2q\). In particular, for each \(k \in \mathbf {N}\) we have \(x_{\tau (\sigma (n))} \in U_k\) for all large \(n \in \mathbf {N}\). Hence
By the arbitrariness of k, it follows that \(\mathfrak {u}(\sigma ,\ell )\ge 2q>q\), that is, \(\sigma \in V(\ell ,q)\).
Conversely, fix \(\ell \in X\), \(\sigma \in \Sigma \), and \(q>0\) such that \(\sigma \in V(\ell ,q)\), hence \(\Vert A_k\Vert _\varphi >q\) for all \(k \in \mathbf {N}\), where \(A_k:=\{n \in \mathbf {N}: x_{\sigma (n)} \in U_k\}\). Let us define recursively a sequence \((F_k)\) of finite subsets of \(\mathbf {N}\) as it follows. Pick \(F_1\subseteq A_1\) such that \(\varphi (F_1) \ge q\) (which is possibile since \(\varphi \) is a lscsm); then, for each integer \(k\ge 2\), let \(F_k\) be a finite subset of \(A_k\) such that \(\min F_k>\max F_{k1}\) and \(\varphi (F_k) \ge q\) (which is possible since \(\Vert A_k \setminus [1,\max F_{k1}]\Vert _\varphi =\Vert A_k\Vert _\varphi >q\)). Let \((y_n)\) be the increasing enumeration of the set \(\bigcup _k F_k\), and define \(\tau \in \Sigma \) such that \(\tau (n)=y_n\) for all n. It follows by construction that
Therefore \(\ell \in \Lambda _{\sigma (x)}(\mathcal {I},q)\subseteq \Lambda _{\sigma (x)}(\mathcal {I})\), which concludes the proof. \(\square \)
Corollary 4.4
With the same hypotheses of Lemma 4.1, \(\tilde{S}(\ell ,q)\) is comeager for each \(\ell \in \mathrm {L}_x\) and \(q \in (0,1)\).
Proof
Fix \(\ell \in \mathrm {L}_x\) and \(q \in (0,1)\). Then \(\tilde{S}(\ell ,q)\) contains \(V(\ell ,q)\) by Lemma 4.3, which is comeager by Lemma 4.2. \(\square \)
Corollary 4.5
With the same hypotheses of Lemma 4.1, \(\tilde{S}(\ell )\) is comeager for each \(\ell \in \mathrm {L}_x\).
Proof
Thanks to [2, Theorem 2.2], we have
Therefore \(\tilde{S}(\ell )\) contains \(\tilde{S}(\ell ,\frac{1}{2})\), which is comeager by Corollary 4.4. \(\square \)
Remark 4.6
All the analogues from Lemma 4.1 up to Corollary 4.5 hold for permutations, the only difference being in the last part of the proof of Lemma 4.2: let us show that
is nowhere dense for all \(k,s \in \mathbf {N}\). To this aim, fix \(\pi _0 \in \Pi \) and \(n_0 \in \mathbf {N}\) which defines the nonempty open set \( G:=\{\pi \in \Pi : \pi \upharpoonright \{1,\ldots ,n_0\}=\pi _0 \upharpoonright \{1,\ldots ,n_0\}\}. \) Set \(n_1:=\max \{n_0+1,s\}\) and let \((y_n)\) be the increasing enumeration of the infinite set \(\{n \in \mathbf {N}: x_n \in U_k\} \setminus \{\pi _0(1),\ldots ,\pi _0(n_0)\}\). Since \(\varphi \) is a lscsm, there exists \(n_2 \in \mathbf {N}\) such that \(\varphi (\{s,s+1,\ldots ,n_2\})>q\). Lastly, let \(G^\prime \) be the set of all \(\pi \in G\) such that \(\pi (n)=y_n\) for all \(n \in \{s,s+1,\ldots ,n_2\}\). We conclude that
Therefore \(G^\prime \) is a nonempty open subset of G which is disjoint from \(\widehat{W}_{k,s}\).
Lastly, we prove Theorem 2.9.
Proof of Theorem 2.9
The implications (L1) \(\implies \) (L2) \(\implies \) (L5) are analogous to the ones in Theorem 2.2, replacing Lemma 3.3 with Corollary 4.5.
(L5) \(\implies \) (L1) Suppose that \(\mathrm {L}_x\ne \emptyset \), otherwise the claim is trivial. Let \(\mathscr {L}\) be a countable dense subset of \(\mathrm {L}_x\), so that \(\mathscr {L}\subseteq \Lambda _{\sigma (x)}(\mathcal {I}, \frac{1}{2})\) for each \(\sigma \in \tilde{S}:=\bigcap _{\ell \in \mathscr {L}}\tilde{S}(\ell , \frac{1}{2})\), which is comeager by Corollary 4.4. Fix \(\sigma \in \tilde{S}\). On the one hand, taking into account (4), we get \(\Lambda _{\sigma (x)}(\mathcal {I}, \frac{1}{2})\subseteq \Lambda _{\sigma (x)}(\mathcal {I}) \subseteq \mathrm {L}_{\sigma (x)}\subseteq \mathrm {L}_x\). On the other hand, since \(\Lambda _{\sigma (x)}(\mathcal {I},\frac{1}{2})\) is closed by Lemma 4.1, we obtain \(\mathrm {L}_x\subseteq \Lambda _{\sigma (x)}(\mathcal {I})\). Therefore \(\tilde{\Sigma }_x(\mathcal {I})\) contains the comeager set \(\tilde{S}\).
The implications (L3) \(\implies \) (L4) \(\implies \) (L5) \(\implies \) (L3) are analogous, taking into account Remark 4.6. \(\square \)
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Acknowledgements
The authors are grateful to the anonymous referee for several accurate and constructive comments which led, in particular, to a simplification of the proof of Lemma 3.3.
Note added in proof
We realized that our main results (i.e., Theorem 2.2 and Theorem 2.9) have been also recently proved also in [30] for the case of subsequences of real sequences, also answering (in this case) the open question at the end of Section 2.
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Balcerzak, M., Leonetti, P. The Baire Category of Subsequences and Permutations which preserve Limit Points. Results Math 75, 171 (2020). https://doi.org/10.1007/s0002502001289y
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DOI: https://doi.org/10.1007/s0002502001289y
Keywords
 Ideal cluster points
 Ideal limit points
 Meager set
 Analytic Pideal
 Ideal convergence
 Subsequences
 Permutations