The Baire Category of Subsequences and Permutations 3

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Moreover, ℓ ∈ X is an I-limit point of x if there exists a subsequence (xnk ) such that

lim xnk = ℓ 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. Indeed, as it follows by [26, 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]. Let also Lx := Γx(Fin) be the set of accumulation points of x, and note that Lx = Λx(Fin) if X is first countable. Hence Λx(I) ⊆ Γx(I) ⊆ Lx. We refer the reader to [26] for characterizations of I-cluster points and [2] for their relation with I-limit points. Lastly, we recall that the sequence x is said to be I-convergent to ℓ ∈ X, shortened as x →I ℓ, if

{n ∈ N : xn ∈/ U} ∈ I for each neighborhood U of ℓ. Assuming that X is ﬁrst countable, it follows by [26, Corollary 3.2] that, if x →I ℓ then Λx(I)=Γx(I) = {ℓ}, provided that I is a P-ideal. In addition, if X is also compact, then x →I ℓ is in fact equivalent to Γx(I)= {ℓ}, even if I is not a P-ideal, cf. [26, Corollary 3.4]. Let Σ be the sets of strictly increasing functions on N, that is, Σ := {σ ∈ NN : ∀n ∈ N, σ(n) < σ(n + 1)}; also, let Π be the sets of permutations of N, that is, Π := {π ∈ NN : π is a bijection}. N Note that both Σ and Π are Gδ-subsets of the Polish space N , hence they are Polish spaces as well by Alexandrov’s theorem; in particular, they are not meager in themselves, cf. [34, Chapter 2]. Given a sequence x and σ ∈ Σ, we denote by σ(x) the subsequence (xσ(n)). Similarly, given π ∈ Π, we write π(x) for the rearranged sequence (xπ(n)). We identify each subsequence of (xkn ) of x with the function σ ∈ Σ deﬁned by σ(n)= kn for all n ∈ N and, similarly, 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 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 2.9). 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). The Baire Category of Subsequences and Permutations 3

2. Main results 2.1. I-cluster points. It has been shown in [25] that, from a topological view- point, 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 : Ac ∈ I}; that is,

Σx(I) := σ ∈ Σ:Γσ(x)(I)=Γx(I) is comeager, cf. also [27] for the case I = 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

Πx(I) := π ∈ Π:Γπ(x)(I)=Γx(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 Fσ-separated from its dual filter I ⋆ if there exists an Fσ-set B ⊆ P(N) such that I ⊆ B and B ∩ I = ∅ (with the language of [9, 20], the filter I⋆ has rank ≤ 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 I be an ideal which is Fσ-separated ⋆ from its dual filter I . Then Σx(I) is not meager if and only if Γx(I)=Lx. Moreover, in this case, it is comeager. As it has been shown in [33, Corollary 1.5], the family of ideals I which are ⋆ Fσ-separated from I includes all Fσδ-ideals. In addition, a Borel ideal is Fσ- separated 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 ∞ {A ⊆ N : ∀ n ∈ N, {a ∈ A : ν2(a)= n} ∈ Fin} where ν2(n) stands for the 2-adic valuation of n), see [21, Theorem 4]. In particular, Fin × Fin is a Fσδσ-ideal which is not Fσ-separated from its dual filter. For related results on Fσ-separation, see [11, Proposition 3.6] and [37]. We show that the analogue of Theorem 2.1 holds for all meager ideals. Hence, this includes new cases as, for instance, I = Fin × Fin. It is worth noting that every meager ideal I is Fσ-separated from the Fréchet ⋆ ⋆ filter Fin (see Proposition 3.1 below), hence I is Fσ-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:

(c1) Σx(I) is comeager in Σ; (c2) Σx(I) is not meager in Σ; (c3) Πx(I) is comeager in Π; (c4) Πx(I) is not meager in Π; (c5) Γx(I)=Lx. 4 M. Balcerzak and P. Leonetti

It is worth to spend some comments on the class C of first countable spaces such that all closed sets are separable. It is clear that separable metric spaces belong to X. However, C contains also nonmetrizable spaces: Example 2.3. It has been shown by Ostaszewski in [30] 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 ∈ 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 C : Example 2.4. Let X be the Sorgenfrey plane, that is, the product of two copies of the real line R endowed with the lower limit topology. It is known that X is first countable and separable. Moreover, the anti-diagonal {(x, −x): x ∈ R} is a closed uncountable discrete subspace of X, hence not separable, cf. [35, pp. 103–105]. Therefore X∈ / C . A similar example is the Moore plane with the tangent disk topology, cf. [35, p. 176]. We can also show that there exists a first countable space outside 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∈ / C . Thanks to Remark 3.4 below, Theorem 2.2 holds if the separability of all closed sets is replaced by the condition that Lx is countable for each sequence x taking values in X. Indeed, the latter is verified because Lx ⊆{xn : n ∈ N}. We leave as open question whether there exists a topological space X (necessar- ily outside C ) and a meager ideal I for which Theorem 2.2 fails. It is well possible that our main result extends beyond 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 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., xn = 1 if n ∈ A and xn = 0 otherwise. Then x →I 0. In particular, Γx(I)= {0}. Note that a subsequence σ(x) is I-convergent to 0 if and only if Γσ(x) = {0}. Then

−1 Σx(I)= {σ ∈ Σ: σ [A] ∈ I}. Considering that σ−1[A] ∪ σ−1[A − 1] is coﬁnite, we have either σ−1[A] ∈ I or σ−1[A − 1] ∈ I. Let T : Σ → Σ be the embedding deﬁned by σ 7→ σ +1, so that The Baire Category of Subsequences and Permutations 5

Σ is homeomorphic to the open set T [Σ] = {σ ∈ Σ: σ(1) ≥ 2}. Notice that −1 T [Σx(I)] = {T (σ): σ ∈ Σx(I)} = {σ +1 ∈ Σ: σ [A] ∈ I} = {σ ∈ Σ: σ−1[A − 1] ∈I}∩ T [Σ], which implies that the open set T [Σ] is contained in Σx(I) ∪ T [Σx(I)]. Therefore both Σx(I) and T [Σx(I)] are not meager. A similar example can be found for Πx(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 noted by the referee, the equivalences (c1) ⇐⇒ (c2) and (c3) ⇐⇒ (c4), and their analogues in the next results, can be viewed a consequence a general topological 0-1 law stating that every tail set with the property of Baire is either meager or comeager, see [31, Theorem 21.4]. However, it seems rather difficult to show that the tail sets Σx(I) and Πx(I) have the property of Baire, provided that I is meager (note that this is surely false if I is a maximal ideal, as it follows by Example 2.6). 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. This is somehow related to [1, Theorem 2.1] and [3, Theorem 1.1]; cf. also [28, Theorem 3] for a measure theoretical non-analogue. Corollary 2.7. 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:

(i1) {σ ∈ Σ: σ(x) →I ℓ} is comeager in Σ; (i2) {σ ∈ Σ: σ(x) →I ℓ} is not meager in Π; (i3) {π ∈ Π: π(x) →I ℓ} is comeager in Σ; (i4) {π ∈ Π: π(x) →I ℓ} is not meager in Π; (i5) limn xn = ℓ. The proofs of Theorem 2.2 and Corollary 2.7 follow in Section 3. 2.2. I-limit points. Given a sequence x and an ideal I, deﬁne

Σ˜ x(I) := {σ ∈ Σ:Λσ(x)(I)=Λx(I)} and its analogue for permutations

Π˜ x(I) := {π ∈ Π:Λπ(x)(I)=Λx(I)}. It has been shown in [25] 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 (µn) of submeasures with ﬁnite and pairwise disjoint supports such that I = {A ⊆ N : limn µn(A)=0}. More precisely: 6 M. Balcerzak and P. Leonetti

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 I be generalized density ideal. Then Σ˜ x(I) is not meager if and only if Λx(I)=Lx. 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 P-ideals. We 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 2.9. 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:

(L1) Σ˜ x(I) is comeager in Σ; (L2) Σ˜ x(I) is not meager in Σ; (L3) Π˜ x(I) is comeager in Π; (L4) Π˜ x(I) is not meager in Π; (L5) Γx(I)=Lx. 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 Section 4. We leave as an open question to check whether Theorem 2.9 may be extended to all meager ideals.

3. Proofs for I-cluster points We start with a characterization of meager ideals (to the best of our knowledge, condition (m3) is novel). Here, a set A ⊆ P(N) is called hereditary if it is closed under subsets. Proposition 3.1. Let I be an ideal on N. Then the following are equivalent: (m1) I is meager; (m2) There exists a strictly increasing sequence (ιn) of positive integers such that A∈ / I whenever N ∩ [ιn, ιn+1) ⊆ A for infinitely many n ∈ N; ⋆ (m3) I is Fσ-separated from the Fréchet filter Fin . Proof. (m1) ⇐⇒ (m2) See [36, Theorem 2.1]; cf. also [4, Theorem 4.1.2]. (m2) =⇒ (m3) Define In := N ∩ [ιn, ιn+1) for all n ∈ N. Then I ⊆ F , where F := Sk Fk and ∀k ∈ N, Fk := \ {A ⊆ N : In 6⊆ A}. (1) n≥k

Note that each Fk is closed and it does not contain any coﬁnite set. Therefore I ⋆ is separated from Fin by the Fσ-set F . The Baire Category of Subsequences and Permutations 7

(m3) =⇒ (m1) Suppose that there exists a sequence (Fk) of closed sets in N ⋆ {0, 1} such that I ⊆ F := Sk Fk and F ∩ Fin = ∅. Then each Fk has empty interior (otherwise it would contain a coﬁnite set). We conclude that I is contained in a countable union of nowhere dense sets.

It is clear that condition (m3) is weaker than the extendability of I to a Fσ- 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 I be a meager ideal. Then {σ ∈ Σ: σ−1[A] ∈/ I} and {π ∈ Π: π−1[A] ∈/ I} are comeager for each infinite set A ⊆ N. Proof. Fix an infinite set A ⊆ N. As in the proof of Proposition 3.1, we can define intervals In := N ∩ [ιn, ιn+1) for all n ∈ N such that a set S ⊆ N does not belong to I whenever S contains infinitely many intervals Ins. At this point, for each n, k ∈ N, define the sets −1 Xk := {σ ∈ Σ: Ik ⊆ σ [A]} and Yn := Sk≥n Xk. Note that each Xk is open and that each Yn is open and dense. Therefore the set −1 Tn Yn, which can be rewritten as {σ ∈ Σ: σ [A] ∈/ I}, is comeager. The proof that {π ∈ Π: π−1[A] ∈/ I} is comeager is analogous. Lemma 3.3. Let x be a sequence in a first countable space X and let I be a meager ideal. Then

S(ℓ) := σ ∈ Σ: ℓ ∈ Γσ(x)(I) and P (ℓ) := π ∈ Π: ℓ ∈ Γπ(x)(I) are comeager for each ℓ ∈ Lx.

Proof. Assume that Lx =6 ∅, otherwise there is nothing to prove. Fix ℓ ∈ Lx and let (Um) be a decreasing local base at ℓ. For each m ∈ N, define the infinite set Am := {n ∈ N : xn ∈ Um}. Thanks to Lemma 3.2, the set Bm := {σ ∈ Σ: −1 σ [Am] ∈/ I} is comeager. Since S(ℓ) can be rewritten as Tm Bm, it follows that S(ℓ) is comeager. The proof that P (ℓ) is comeager is analogous. We are finally ready to prove Theorem 2.2. Proof of Theorem 2.2. (c1) =⇒ (c2) It is obvious. (c2) =⇒ (c5) Suppose that there exists ℓ ∈ Lx \ Γx(I). Then Σx(I) is contained in Σ \ S(ℓ), which is meager by Lemma 3.3. (c5) =⇒ (c1) Suppose that Lx =6 ∅, otherwise the claim is trivial. Let L be a L countable dense subset of Lx, so that ⊆ Γσ(x)(I) for each σ ∈ S := Tℓ∈L S(ℓ), which is comeager by Lemma 3.3. Fix σ ∈ S. On the one hand, Γσ(x)(I) ⊆ Lσ(x) ⊆ Lx. On the other hand, since Γσ(x)(I) is closed by [26, Lemma 3.1(iv)], we get Lx ⊆ Γσ(x)(I). Therefore S ⊆ Σx(I). The implications (c3) =⇒ (c4) =⇒ (c5) =⇒ (c3) are analogous. 8 M. Balcerzak and P. Leonetti

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, Lx is countable. Lemma 3.5. Let I be an ideal and x be a sequence in a ﬁrst countable compact space. Then x →I ℓ if and only if Γx(I)= {ℓ}. Proof. It follows by [26, Corollary 3.4]. Proof of Corollary 2.7. (i1) =⇒ (i2) It is obvious. (i2) =⇒ (i5) By Lemma 3.5, the hypothesis can be rewritten as Γx(I)= {ℓ}. Hence, condition (i2) is equivalent to the nonmeagerness of {σ ∈ Σ:Γσ(x) = Γx(I)= {ℓ}}. The claim follows by Theorem 2.2 and Remark 3.4. (i5) =⇒ (i1) If x → ℓ then σ(x) →I ℓ for all σ ∈ Σ. The implications (i3) =⇒ (i4) =⇒ (i5) =⇒ (i3) are analogous.

4. Proofs for I-limit points A lower semicontinuous submeasure (in short, lscsm) is a monotone subadditive function ϕ : P(N) → [0, ∞] such that ϕ(∅) = 0, ϕ(F ) < ∞ for all F ∈ Fin, and ϕ(A) = limn ϕ(A ∩ [1, n]) for all A ⊆ N. By a classical result of Solecki, an ideal I is an analytic P-ideal if and only if there exists a lscsm ϕ such that

I = Exh(ϕ) := {A ⊆ N : kAkϕ =0} and 0 < kNkϕ ≤ ϕ(N) < ∞, (2) where kAkϕ := limn ϕ(A \ [1, n]) for all A ⊆ N, see [32, Theorem 3.1]. Note that k·kϕ is a submeasure which is invariant modulo finite sets. Moreover, replacing ϕ with ϕ/kNkϕ in (2), we can assume without loss of generality that kNkϕ = 1. Given a sequence x in a first countable topological space X and an analytic P-ideal I = Exh(ϕ), we define the function

u : Σ × X → R :(σ, ℓ) 7→ lim k{n ∈ N : xσ(n) ∈ Uk}kϕ, (3) k→∞ where (Uk) is a decreasing local base of neighborhoods at ℓ ∈ X. Clearly, the limit in (3) exists and it is independent of the choice of (Uk). Lemma 4.1. Let x be a sequence in a ﬁrst countable space X and let I = Exh(ϕ) be an analytic P-ideal. Then, the section u(σ, ·) is upper semicontinuous for each σ ∈ Σ. In particular, the set

Λσ(x)(I, q) := {ℓ ∈ X : u(σ, ℓ) ≥ q} is closed for all q > 0. Proof. See [2, Lemma 2.1]. Lemma 4.2. With the same hypotheses of Lemma 4.1, the set V (ℓ, q) := {σ ∈ Σ: u(σ, ℓ) > q} is either comeager or empty for each ℓ ∈ X and q ∈ (0, 1). The Baire Category of Subsequences and Permutations 9

Proof. Suppose that V (ℓ, q) =6 ∅, so that ℓ ∈ Lx, and note that

Σ \ V (ℓ, q)= [{σ ∈ Σ: k{n ∈ N : xσ(n) ∈ Uk}kϕ ≤ q} k≥1

= [{σ ∈ Σ : lim sup ϕ({n ≥ t : xσ(n) ∈ Uk}) ≤ q} t→∞ k≥1

= [ [ \{σ ∈ Σ: ϕ({n ≥ t : xσ(n) ∈ Uk}) ≤ q}. k≥1 s≥1 t≥s Then, it is suﬃcient to show that

Wk,s := \{σ ∈ Σ: ϕ({n ≥ t : xσ(n) ∈ Uk}) ≤ q} t≥s is nowhere dense for all k,s ∈ N. To this aim, for every nonempty open set Z ⊆ Σ, we need to prove that there exists a nonempty open subset S ⊆ Z such that S ∩ Wk,s = ∅. Fix a nonempty open set Z ⊆ Σ and σ0 ∈ Z so that there exists n0 ∈ N for which ′ Z := {σ ∈ Σ: σ ↾ {1,...,n0} = σ0 ↾ {1,...,n0}} ⊆ Z. ′ Since ℓ ∈ Lx, there exists σ1 ∈ Z such that limn xσ1(n) = ℓ. Therefore

ϕ({n ≥ n1 : xσ1(n) ∈ Uk}) ≥ k{n ≥ n1 : xσ1(n) ∈ Uk}kϕ u = k{n ∈ N : xσ1(n) ∈ Uk}kϕ = (σ1,ℓ)=1, where n1 := max{n0 +1,s}. At this point, since ϕ is a lscsm, it follows that there exists an integer n2 > n1 such that ϕ({n ∈ N ∩ [n1, n2]: xσ1(n) ∈ Uk}) > q. ′ Therefore S := {σ ∈ Z : σ ↾ {n1,...,n2} = σ1 ↾ {n1,...,n2}} is a nonempty open set contained in Z and disjoint from Wk,s. Indeed

∀σ ∈ S, ϕ({n ≥ s : xσ(n) ∈ Uk}) ≥ ϕ({n ∈ N ∩ [n1, n2]: xσ(n) ∈ Uk}) > q by the monotonicity of ϕ. Lemma 4.3. With the same hypotheses of Lemma 4.1, we have

∀ℓ ∈ X, {σ ∈ Σ: ℓ ∈ Λσ(x)(I)} = Sq>0 V (ℓ, q).

In addition, S˜(ℓ, q) := {σ ∈ Σ: ℓ ∈ Λσ(x)(I, q)} contains V (ℓ, q). Proof. Fix ℓ ∈ X and σ ∈ S˜(ℓ), where

S˜(ℓ) := {σ ∈ Σ: ℓ ∈ Λσ(x)(I)}.

Then there exist τ ∈ Σ and q > 0 such that limn xτ(σ(n)) = ℓ and kτ(N)kϕ ≥ 2q. In particular, for each k ∈ N we have xτ(σ(n)) ∈ Uk for all large n ∈ N. Hence

k{n ∈ N : xσ(n) ∈ Uk}kϕ ≥ k{n ∈ N : xσ(n) ∈ Uk} ∩ τ(N)kϕ = kτ(N)kϕ ≥ 2q. By the arbitrariness of k, it follows that u(σ, ℓ) ≥ 2q > q, that is, σ ∈ V (ℓ, q). 10 M. Balcerzak and P. Leonetti

Conversely, fix ℓ ∈ X, σ ∈ Σ, and q > 0 such that σ ∈ V (ℓ, q), hence kAkkϕ > q for all k ∈ N, where Ak := {n ∈ N : xσ(n) ∈ Uk}. Let us define recursively a sequence (Fk) of finite subsets of N as it follows. Pick F1 ⊆ A1 such that ϕ(F1) ≥ q (which is possibile since ϕ is a lscsm); then, for each integer k ≥ 2, let Fk be a finite subset of Ak such that min Fk > max Fk−1 and ϕ(Fk) ≥ q (which is possible since kAk \ [1, max Fk−1]kϕ = kAkkϕ > q). Let (yn) be the increasing enumeration of the set Sk Fk, and define τ ∈ Σ such that τ(n)= yn for all n. It follows by construction that

lim xτ(σ(n)) = ℓ and kτ(N)kϕ ≥ lim inf ϕ(Fk) ≥ q > 0. n→∞ k→∞

Therefore ℓ ∈ Λσ(x)(I, q) ⊆ Λσ(x)(I), which concludes the proof. Corollary 4.4. With the same hypotheses of Lemma 4.1, S˜(ℓ, q) is comeager for each ℓ ∈ Lx and q ∈ (0, 1).

Proof. Fix ℓ ∈ Lx and q ∈ (0, 1). Then S˜(ℓ, q) contains V (ℓ, q) by Lemma 4.3, which is comeager by Lemma 4.2. Corollary 4.5. With the same hypotheses of Lemma 4.1, S˜(ℓ) is comeager for each ℓ ∈ Lx. Proof. Thanks to [2, Theorem 2.2], we have

Λσ(x)(I)= Sq>0 Λσ(x)(I, q). (4)

Therefore S˜(ℓ) contains S˜(ℓ, 1/2), which is comeager by Corollary 4.4. 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

Wk,s := [{π ∈ Π: ϕ({n ≥ t : xπ(n) ∈ Uk}) ≤ q} c t≥s is nowhere dense for all k,s ∈ N. To this aim, fix π0 ∈ Π and n0 ∈ N which defines the nonempty open set G := {π ∈ Π: π ↾ {1,...,n0} = π0 ↾ {1,...,n0}}. Set n1 := max{n0 +1,s} and let (yn) be the increasing enumeration of the infinite set {n ∈ N : xn ∈ Uk}\{π0(1),...,π0(n0)}. Since ϕ is a lscsm, there exists ′ n2 ∈ N such that ϕ({s,s +1,...,n2}) > q. Lastly, let G be the set of all π ∈ G such that π(n)= yn for all n ∈{s,s +1,...,n2}. We conclude that ′ ∀π ∈ G , ϕ({n ≥ s : xπ(n) ∈ Uk}) ≥ ϕ({s,s +1,...,n2}) > q. ′ Therefore G is a nonempty open subset of G which is disjoint from Wk,s. c Lastly, we prove Theorem 2.9. Proof of Theorem 2.9. The implications (L1) =⇒ (L2) =⇒ (L5) are analogous to the ones in Theorem 2.2, replacing Lemma 3.3 with Corollary 4.5. The Baire Category of Subsequences and Permutations 11

(L5) =⇒ (L1) Suppose that Lx =6 ∅, otherwise the claim is trivial. Let L 1 be a countable dense subset of Lx, so that L ⊆ Λσ(x)(I, /2) for each σ ∈ S˜ := ˜ 1 ˜ Tℓ∈L S(ℓ, /2), which is comeager by Corollary 4.4. Fix σ ∈ S. On the one hand, 1 taking into account (4), we get Λσ(x)(I, /2) ⊆ Λσ(x)(I) ⊆ Lσ(x) ⊆ Lx. On the 1 other hand, since Λσ(x)(I, /2) is closed by Lemma 4.1, we obtain Lx ⊆ Λσ(x)(I). Therefore Σ˜ x(I) contains the comeager set S˜. The implications (L3) =⇒ (L4) =⇒ (L5) =⇒ (L3) are analogous, taking into account Remark 4.6.

Acknowledgments. The authors are grateful to the anonymous referee for sev- eral accurate and constructive comments which led, in particular, to a simpliﬁca- tion of the proof of Lemma 3.3.

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Institute of Mathematics, Lodz University of Technology, ul. Wolcza´ nska´ 215, 93-005 Lodz, Poland E-mail address: [email protected] Department of Decision Sciences, Universita` “Luigi Bocconi”, via Roentgen 1, 20136 Milan, Italy E-mail address: [email protected] URL: https://sites.google.com/site/leonettipaolo/