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Products of Menger spaces: a combinatorial approach

We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.

in the real line, or even just metrizable examples [23,Problem 6.7]. This problem, proposed by Scheepers long ago, resisted tremendous efforts thus far.
For brevity, sets of real numbers are called here real sets. Assuming the Continuum Hypothesis, there are two Luzin sets whose product is not Menger [12,Theorem 3.7]. An uncountable real set is Luzin if its intersection with every meager (Baire first category) set is countable. An uncountable real set X is concentrated if it has a countable subset D such that the set X \ U is countable for every open set U containing D. Every Luzin set is concentrated, and every concentrated set has Menger's property. This approach extends to obtain similar examples using a set theoretic hypothesis about the meager sets that is weaker than the Continuum Hypothesis [20,Theorem 49]. Later methods [27,Theorem 9.1] were combined with reasoning on meager sets to obtain examples using another portion of the Continuum Hypothesis [19,Theorem 3.3]. Here, we introduce a purely combinatorial approach to this problem. We obtain examples using hypotheses milder than earlier ones, as well as examples using hypotheses that are incompatible with the Continuum Hypothesis. To this end, we introduce the key notion of bi-d-unbounded set, and determine the limits on its possible existence. We extend these results to variations of Menger's property parameterized by filters and semifilters. For a semifilter S, we introduce the notion of S-scale. These scales provably exist, and capture a number of distinct special cases used in earlier works.
The second part of the paper, beginning with Section 5, establishes provably productive properties among semifilter-parameterized Menger properties. If S is an ultrafilter, then every S-scale gives rise to a productively S-Menger space. We deduce that each of these variations of Menger's property is strictly between Hurewicz's and Menger's classic properties.
The last section includes a discussion of related results and open problems suggested by our results.

Products of Menger spaces
Towards a combinatorial treatment of the questions discussed here, we identify the Cantor space {0, 1} N with the family P(N) of all subsets of the set N. Since the Cantor space is homeomorphic to Cantor's real set, every subspace of the space P(N) is considered as a real set.
The space P(N) splits into two important subspaces: the family of infinite subsets of N, denoted [N] ∞ , and the family of finite subsets of N, denoted [N] <∞ . We identify every set a ∈ [N] ∞ with its increasing enumeration, an element of the Baire space N N . Thus, for a natural number n, a(n) is the n-th element in the increasing enumeration of the set a. This way, we have [N] ∞ ⊆ N N , and the topology of the space [N] ∞ (a subspace of the Cantor space P(N)) coincides with the subspace topology induced by N N . This explains some of the elementary assertions made here; moreover, notions defined here for [N] ∞ are often adaptations of classic notions for N N . Depending on the interpretation, points of the space [N] ∞ are referred to as sets or functions.
For functions a, b ∈ [N] ∞ , we write a ≤ b if a(n) ≤ b(n) for all natural numbers n, and a ≤ * b if a(n) ≤ b(n) for almost all natural numbers n, that is, the set of exceptions { n : b(n) < a(n) } is finite. We follow the convention that bounded means has an upper bound in the ambient superset.
the cardinality of every ≤-bounded subset of the set X is smaller than κ.
Remark 2.2. For cardinal numbers κ of uncountable cofinality, which will be the case in the present paper, the notion of κ-unbounded defined here is equivalent to its variation using the relation ≤ * instead of ≤. This is not the case for cardinal numbers of countable cofinality.
This convention applies to all binary relations.
There are, provably, d-unbounded sets and cf(d)-unbounded sets:
There is a natural number n such that and thus a ⊎ d ′ b. Therefore, By Lemma 2.3, the real set is d-concentrated on the set [N] <∞ . In particular, the set Y is Menger.
For sets a, b ∈ P(N), let a ⊕ b denote the symmetric difference of the sets a and b. With respect to the operator ⊕, the space P(N) is a topological group.
For each function d in the dominating set D we started with, let a ∈ A be a function such that d ∈ I a . As a ∈ X and a ⊎ d ′ ∈ Y , we have Let X be a real set of cardinality smaller than d. Then the set X is trivially Menger: the topology used is irrelevant, as long as we restrict attention to countable covers. In particular, all finite powers of the real set X are Menger, even for countable Borel covers; a strong property [20].  The most important application of Theorem 2.7 appears in the next section.

Bi-d-unbounded sets
For a set a ∈ P(N), let a c : The existence of bi-d-unbounded sets and bi-cf(d)-unbounded sets is a mild hypothesis.
We use the following lemma, to which we provide a short, direct proof.
We may assume that 1 / ∈ x for all sets x ∈ X. Since |X| < d, there is a function a ∈ [N] ∞ such that the sets Fix a set x ∈ X. Let n be a member of the infinite set r ∩ I x . There are at least two elements in the set [a(n), a(n + 1)) ∩x; letx(i) be the minimal one. Then x(x(i)) = x(i + 1) ∈ [a(n), a(n + 1)). Since n ∈ r, the set b c ∩ [a(n), a(n + 1)) is empty, and thus a(n We prove that the family A is reapable. We may assume that for each set a ∈ A and each finite set s, we have a \ s ∈ A.
Since the set X is bi-d-unbounded, the set is a union of less than d sets, each of cardinality smaller than d. Thus, there is an element r ∈ X that is not included in that set, that is, such that A < ∞ r, r c . The set r reaps the family A: Fix a set a ∈ A. Assume that the set a ∩ r is finite. Then the set a ′ := a \ r is in A, and thus a ′ < ∞ r c . But a ′ ⊆ r c , and thus r c ≤ a ′ ; a contradiction. For the same reason, the set a \ r is infinite, too.
(3) ⇒ (1): If the cardinal number d is regular, then the previously established implication applies. And if it is singular, then as explained in the previous implication, we have d ≤ r.
In either case, we are done.
Every real set of cardinality smaller than cov(M) is Rothberger [12,Theorem 4.2], and therefore so is every cov(M)concentrated real set. Since cov(M) ≤ r [7, Theorem 5.19], we obtain the following result.  3 Important examples of semifilters include the maximal semifilter [N] ∞ , the minimal semifilter cF of all cofinite sets, and every nonprincipal ultrafilter on N.

Filter-Menger spaces
Let S be a semifilter.
The κ-unbounded sets presented in the previous sections are instances of the following notion, which generalizes the earlier notion of b(S)-scale [27, Definition 2.8].
of an element of S + and an element of S.
For the filter cF of cofinite sets, the property cF-Menger is the classic Hurewicz property [11]. Thus, for every semifilter S, we have the following implications.  By filter we mean a semifilter closed under finite intersections. If F is a filter, then a ∩ b ∈ F + for all sets a ∈ F and b ∈ F + . And if F is an ultrafilter, then F + = F .  (1) The semifilter S is nonmeager. The following theorem implies that this assertion cannot be established for spaces that are not Hurewicz.
Let P be a property of topological spaces. A real set X is productively P if for each topological space Y with the property P, the product space X × Y has the property P. The question whether productively P implies productively Q, for P and Q covering properties among those studied here, has a long history. The remainder of this paragraph assumes that d = ℵ 1 . Aurichi and Tall [3] improved several earlier results by proving that every productively Lindelöf space is Hurewicz. It was later shown that every productively Lindelöf space is productively Hurewicz and productively Menger [18,Theorem 8.2]. Thus, productively Lindelöf implies productively Menger, and the following theorem shows that productively Menger suffices to imply productively Hurewicz. (  (1), there is a d-concentrated real set Y such that the planar set Z × Y is not Menger. Since the set Z × Y is an upper continuous image of the product space X × H × Y , the latter space is not Menger, too. As the space H is Hurewicz and hereditarily Lindelöf and the set Y is d-concentrated, the product space H × Y is Menger [28,Theorem 4.6]. In summary, the product of the real set X and the Menger, hereditarily Lindelöf space H × Y is not Menger.
Some special hypothesis is necessary for Theorem 4.8: The union of less than g Menger real sets is Menger [29,26]. Assume that b < g. Then any unbounded real set X ⊆ [N] ∞ of cardinality b is productively Menger but not Hurewicz.

Productive real sets
In this section, we establish preservation of some properties under products. We begin with a generalization of an earlier result [18, Lemma 6.3] to general topological spaces. The earlier proof [18, Lemma 6.3] does not apply in this general setting; we provide an alternative proof.  1, a y (k)))×{y} ⊆ U n m . Let a y (k +1) be the minimal natural number such that a y (k+1) ≥ m and P([a y (k), a y (k+1)) c )×{y} ⊆ U n m . Since our open covers are ascending, we have P([a y (k), a y (k + 1)) c ) × {y} ⊆ U n ay(k+1) . Notice that the number a y (k + 1) is minimal with this property. As the set P([a y (k), a y (k + 1)) c ) × {y} is compact, there is an open neighborhood V y k+1 ⊆ V y k of the point y such that P([a y (k), a y (k + 1)) c ) × V y k+1 ⊆ U n ay(k+1) . Define Φ(y)(n) := a y (n + 1). For each point y ′ ∈ V y n+1 , we have y ′ ∈ V y k for all k = 1, . . . , n + 1. The sequence a y ′ (1), . . . , a y ′ (n + 1) is bounded by the sequence a y (1), . . . , a y (n + 1), coordinate-wise: For k = 1, we have a y ′ (1) = a y (1) = 1. Assume that a y ′ (k) ≤ a y (k). Then , and, by the minimality of the number a y ′ (k + 1), we have a y ′ (k + 1) ≤ a y (k + 1), too.
For filters, we obtain a productive version of Proposition 4.5.

Theorem 5.2. Let F be a filter and X ⊆ [N] ∞ be an F -scale. For each F -Menger space Y , every upper continuous image of the product space
, an element of the semifilter F + . Then the cardinality of the set This Proof. Every product of a metrizable Lindelöf space and a hereditarily Lindelöf spaces is Lindelöf. Apply Theorem 5.2.
The following theorem was previously known for b-scales, a special kind of cF-scales [18,Theorem 6.5]. This theorem and the subsequent one improve upon earlier results [27, Proof. Let Y be a hereditarily Lindelöf, Hurewicz space. Since the space Y is hereditarily Lindelöf, the product space ( for all natural numbers n and for k ∈ [h(n), h(n + 1)). Then Ψ[Y ] ≤ * b ≤b.
Since the set X is an S-scale, there is a function c ∈ [N] ∞ such thatb ≤ S + c and all but less than b functions x ∈ X belong to the set Proof. Fix a function x ∈X. Then [c ≤ x] ∈ S, and thus the set [c ≤ x] ∩ [h(n), h(n + 1)) is nonempty for almost all natural numbers n. Let Then, for almost all natural numbers n ∈ d, there are natural numbers x(h(n + 1)).  (1) There is a cofinal F -scale.
Proof. (1) ⇒ (2): Since F is a filter, we have F ⊆ F + , and thus b(F ) ≤ b(F + ). Apply Proposition 6.2(3).  A superfilter (also called grille or coideal) is a semifilter S such that a∪b ∈ S implies a ∈ S or b ∈ S. A semifilter S is a superfilter if and only if the semifilter S + is a filter. Equivalently, a superfilter is a union of a family of ultrafilters. For example, the set [N] ∞ = cF + is a superfilter.

Proposition 6.6. Let S be a superfilter. A set X ⊆ [N] ∞ is a cofinal S-scale if and only if it is an S-scale.
The proof of the following assertion is similar to that of Proposition 4.5. Let U be an ultrafilter, and X ⊆ [N] ∞ be a U-scale. By Proposition 6.6, the set X is in fact a cofinal U-scale. Using Proposition 6.7, we obtain an alternative derivation of Corollary 4.6. Similarly, Theorem 6.4 generalizes Theorem 5.3 (2). Theorem 6.4 cannot be extended to all semifilters, and not even to all superfilters: By Theorems 3.2-3.3, the hypothesis d ≤ r implies that Theorem 6.4 does not hold for the superfilter [N] ∞ . The latter assertion also follows from the following theorem that is, in fact, established by the proof of Theorem 4.7.
Theorem 6.8. Assume that b = d. Let S be a semifilter. The following assertions are equivalent: (1) The semifilter S is nonmeager. Proof of (1). By Theorem 6.8(3) and Theorem 6.4, using that products of Hurewicz sets and d-concentrated real sets are Menger [28, Theorem 4.6].

Comments and open problems
We restrict attention to real sets throughout this section, except for the last subsection. The Menger productivity problem, whether Menger's property is consistently preserved by finite products, remains open. The hypothesis d ≤ r provides two Menger sets whose product is not Menger (Theorems 2.7 and 3.2). It is well known that this immediately provides a Menger set whose square is not Menger. Indeed, assume that X and Y are Menger real sets such that the planar set X × Y is not Menger. The set X ∪ Y is Menger. We may assume that X ⊆ (0, 1) and Y ⊆ (2, 3). Then the set X ×Y is a closed subset of the square (X ∪Y ) 2 . Menger's property is hereditary for closed subsets. 7.1. A combinatorial characterization of the cardinal number min{r, d}. Aubrey [1] proved that min{d, u} ≤ r. Since r ≤ u, the hypothesis d ≤ r in Theorem 3.3 is equivalent to the hypothesis d ≤ u.
Initially, we proved Theorem 3.3 using a stronger hypothesis.
We observed that max{b, cov(M)} ≤ bidi ≤ min{r, d}, and needed that bidi = d to carry out our construction. It is immediate that bidi ≤ d, and the argument in the proof of the implication (2) ⇒ (1) in Theorem 3.3 shows that bidi ≤ r. Answering our question, Mejía pointed out to us that, by a result of Kamburelis and Węglorz, our upper bound on the cardinal number bidi is tight [16] (see Lemma 3.4). We thus have the following characterization of min{r, d}. Proposition 7.2. bidi = min{r, d}.

7.2.
Which κ-unbounded sets are not productively Menger? There are, in ZFC, bunbounded sets (e.g., by Proposition 4.2). Every union of less than max{b, g} Menger sets is Menger [29,26]. Since the hypothesis b < g is consistent, Theorem 2.7 and Corollary 2.11 do not extend to the case κ = b, or to any cardinal number that is consistently smaller than max{b, g}.
For a κ-unbounded set, which we may assume to have cardinality κ, the present proof of Theorem 2.7 requires a partition d = α<κ I α such that for each set J ⊆ κ with |J| < d, we have | α∈J I α | < d. It is not difficult to see that this implies that κ ∈ {cf(d), d}. To this end, it is important to find mild hypotheses implying that there are two Hurewicz sets whose product is not Hurewicz. In the notation of Section 4, Menger's property is [N] ∞ -Menger, and Hurewicz's property is cF-Menger. By Theorems 3.2 and 3.3, [N] ∞scales need not be productively [N] ∞ -Menger. In contrast, by Theorem 5.4, cF-scales are productively cF-Menger. Thus, modern constructions of Hurewicz sets do not help in regard to Problem 7.4. The classic example of a Hurewicz set (if not counting σ-compact sets, which are productively Hurewicz) is a Sierpiński set [24]. If b = cov(N ) = cof(N ), then there is a b-Sierpiński set (which is Hurewicz) whose square is not Hurewicz [20,Theorem 43]. No more general constructions are known.

Problem 7.5. Does the Continuum Hypothesis imply that no Sierpiński set is productively Hurewicz? Productively Menger?
By Theorem 4.8, if b = d, then every productively Menger set is productively Hurewicz. By the discussion following that theorem, if b < g then there are productively Menger sets that are not Hurewicz. Thus meager semifilters do not have the property in Problem 7.7. By Theorem 6.4 and Proposition 6.6, ultrafilters are also not in that category. But the full semifilter [N] ∞ is in this category, by Theorem 6.8.
A b-scale [24] is a particularly simple kind of a cF-scale, for cF the filter of cofinite sets. 7.4. Strictly unbounded sets. Say that a set X ⊆ [N] ∞ is strictly unbounded if for every set A ⊆ [N] ∞ of cardinality smaller than d there is a function x ∈ X such that A ≤ ∞ x. Let X ⊆ [N] ∞ be a strictly unbounded set. By the argument in the proof of Theorem 4.8(1), the set X contains a d-unbounded set. By Theorem 2.7, there is a d-concentrated set Y such that the planar set X × Y is not Menger. If b = d, then every unbounded set in [N] ∞ is strictly unbounded. We thus obtain a generalization of Theorem 4.8.
The construction in Theorem 3.3 that provides a Menger set that is not productively Menger provides, in fact, a Menger strictly unbounded set.
A negative answer to the second item of the following problem implies a negative solution for the Menger productivity problem. Theorem 5.4 provides a positive answer for hereditarily Lindelöf spaces Y . But this restriction is only needed for deducing that the product space (X ∪[N] <∞ ) ×Y is Lindelöf (and similarly for the other results in Section 4). A positive solution for the following problem would suffice. Problem 7.11. Let X be a real set of cardinality smaller than b, and Y be a Hurewicz space. Is the product space X × Y necessarily Lindelöf?