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A characterization of the Menger property by means of ultrafilter convergence

We characterize various Menger/Rothberger related properties by means of ultrafilter convergence, and discuss their behavior with respect to products.

Motivated by classical arguments in the theory of ultrafilter convergence, we give a characterization of the Menger property and of the Rothberger property by means of ultrafilters and filters, respectively. In this vein, we discuss the behavior with respect to products of the above notions, and of some weaker variants.
A summary of the paper follows. In Section 1 we briefly recall the notion of filter convergence, together with some classical examples, which furnish the main motivation for the present paper.
In Section 2 we show that the Menger property allows a characterization in terms of ultrafilter convergence. Actually, our methods work also for the notion in which we fix a bound for the cardinality of the covers under consideration. Even more generally, we can also consider more than countably many families of covers, and even allow infinite subsets to be selected. We are thus led to consider a generalized Rothberger notion R(λ, µ; <κ) which depends on three cardinals, and generalizes simultaneously the Menger property, the Rothberger property, the Menger and the Rothberger properties for countable covers, as well as [κ, µ]-compactness (in particular, countable compactness, initial µ-compactness, Lindelöfness and final κ-compactness). See Definition 2.1.
In Section 3 we study preservation and non preservation under products of the generalized Rothberger notion R(λ, µ; <κ), establishing a strong connection with preservation/non preservation of [κ, µ]-compactness. For example, we get that every product of members of some family F of topological spaces satisfies the Menger property for countable covers if and only if every product of members of F is countably compact.
Finally, in Section 4 we deal with particular cases strongly resembling the Rothberger property (in the sense that only one element can be selected from each cover). In this case, the characterization in terms of convergence involves filters which are not maximal, and this fact can be seen as the one "responsible" for non preservation under products.
The sections of the paper are rather independent and, apart from some comments and with the exceptions mentioned below, can be read in any order. All sections use the (classical) definitions recalled in the first two paragraphs of Section 1. Sections 2-4 rely on Definition 2.1. Finally, Section 4 depends on Lemma 2.2 and Proposition 3.1.

Some facts about ultrafilter convergence
Many topological properties have been characterized in terms of ultrafilter convergence. Recall that if I is a set, (x i ) i∈I is an I-indexed sequence of elements of some topological space X, and F is a filter over I, then a point x ∈ X is said to be an F -limit point of the sequence (x i ) i∈I if {i ∈ I | x i ∈ U} ∈ F , for every open neighborhood U of x (Choquet [Ch, Section IV], Katetov [Ka]). Notice that if X is T 2 , then a sequence has at most one F -limit point; this is the reason for the alternative expression F -convergent sequence. In this note we shall not actually need unicity of F -limits, therefore we shall assume no separation axiom.
Among the many properties which have been considered, definable in terms of ultrafilter convergence, a classical one asks that every sequence has an F -limit point, for some given F . In details, a topological space X is F -compact, for some filter F over I, if every I-indexed sequence of elements of X has some limit point in X (Bernstein [B] for ultrafilters over I = ω, strongly motivated by Robinson non-standard analysis [R]; Saks [Sa] for ultrafilters over cardinals). F -compactness has the pleasant property of being preserved under taking products.
Interesting topological properties arise from conditions asserting that a topological space is D-compact, for all ultrafilters D in some particular class of ultrafilters. For example, a regular topological space X is λ-bounded (i. e., the closure of every subset of cardinality ≤ λ is compact) if and only if X is D-compact, for every ultrafilter D over λ. In particular, a topological space is compact if and only if it is D-compact, for every ultrafilter D. See [Sa,Theorem 2.9, and Section 5].
A much broader range of applications of ultrafilter convergence has been discovered shortly after the appearance of Bernstein paper. For example, it follows from Ginsburg and Saks [GS,p. 404] that a topological space is countably compact if and only if, for every sequence (x i ) i∈ω of elements of X, there exists some ultrafilter D uniform over ω such that the sequence has some D-limit point in X. Here the ultrafilter is not fixed in advance, but, in general, it depends on the sequence. The above characterization of countable compactness is the key to the result, also due to Ginsburg and Saks [GS,Theorem 2.6], that all powers of some space X are countably compact if and only if X is D-compact, for some ultrafilter D uniform over ω. Similar characterizations, and product theorems as well, appear in [Sa] for general types of accumulation properties, including, in particular, initial µcompactness, and, in equivalent form, [µ, µ]-compactness, for µ regular. Saks' results have been subsequently extended to [κ, µ]-compactness by Caicedo [Ca]. Caicedo's treatment has also the advantage of using a single ultrafilter, rather than a family of ultrafilters.
A comprehensive survey of earlier results on the subject can be found in Vaughan [V] and Stephenson [St], together with many related notions and results. See Kočinac and García-Ferreira [GK, in particular, Section 3] for a survey of additional results, and Lipparini [L4, L5, L6] for even more general treatments of the subject.

The Menger property in terms of ultrafilter convergence
In this section we present a characterization of the Menger property along the lines described in the above section. See, e. g., Scheepers [Sc], Tsaban [Ts] and Kočinac [Ko] for information and references about the Menger and related properties.
Our proofs become cleaner if we explicitly settle the cardinalities of the families of covers under consideration. Thus we are lead to consider the following general property which depends on 3 cardinal parameters (and which might also have independent interest).
For cardinals λ, µ and κ, we let [µ] <κ denote the set of all subsets of µ of cardinality <κ, and we let λ ([µ] <κ ) denote the set of all functions from λ to [µ] <κ . To avoid complex formulas in subscripts, we sometimes shall denote a sequence (x i ) i∈I as x i | i ∈ I .
Lemma 2.2. For every topological space X, and λ, µ and κ nonzero cardinals, the following conditions are equivalent.
(2) For every sequence (C α ) α∈λ of families of closed sets of X, each family having cardinality ≤ µ, if, for every choice of subfamilies D α ⊆ C α in such a way that each D α has cardinality < κ, it happens that {C | C ∈ D α , α ∈ λ} = ∅, then there isᾱ ∈ λ such that Cᾱ = ∅.
Proof. The equivalence of (1) and (2) is trivial by taking complements.
We say that an ultrafilter D over λ ( Theorem 2.3. Suppose that µ and κ are infinite cardinals, λ is a nonzero cardinal, and let I = λ ([µ] <κ ). Then, for every topological space X, the following conditions are equivalent.
Suppose that (1) holds, and let the sequence x f | f ∈ I be given. Since µ is infinite, then |[µ] <ω | = µ, thus |C α | ≤ µ, for every α ∈ λ. Suppose that, for every α ∈ λ, D α ⊆ C α , and |D α | < κ. This means that, for every α ∈ λ, there is Z α ⊆ [µ] <ω such that |Z α | < κ, and Notice that | Z α | < κ, since |Z α | < κ, κ is infinite, and each member of |Z α | is finite. If α ∈ λ, and s ∈ Z α , then x f belongs to C α,s , thus x f ∈ {C | C ∈ D α , α ∈ λ}, hence this last set is nonempty. Since, by assumption, X satisfies R(λ, µ; <κ), then, by Lemma 2.2(2), there isᾱ ∈ λ such that Cᾱ = ∅. Thus (2) is proved. Now suppose that (2) holds, and that x f | f ∈ I is a sequence of elements of X. For theᾱ given by (2), we can pick x ∈ Cᾱ. Thus x ∈ Cᾱ ,s , for every s ∈ [µ] <ω . For any neighborhood U of x in X, let where, as in the definition of functional regularity, for every β ∈ µ, we let Aᾱ ,β = {f ∈ I | β ∈ f (ᾱ)}. We want to show that F ∪ G has the finite intersection property. Notice that Hence, in order to prove that F ∪ G has the finite intersection property, it is enough to show that B U ∩ Aᾱ ,s = ∅, for every neighborhood U of x, and every s ∈ [µ] <ω , where we have put Aᾱ ,s = {f ∈ I | s ⊆ f (ᾱ)}. Indeed, for every U and s as above, since x ∈ Cᾱ ,s , there is f ∈ I such that s ⊆ f (ᾱ), and x f ∈ U, and this means exactly that x f belongs to B U ∩ Aᾱ ,s , thus this set is not empty. We have proved that F ∪ G has the finite intersection property.
Thus F ∪ G can be extended to an ultrafilter D over I. Since D ⊇ G, then D is functionally regular. Since D ⊇ F , then x is a D-limit point of x f | f ∈ I . The implication (2) ⇒ (3) is thus proved. Now assume that (3) holds. We shall prove that Condition (3) in Lemma 2.2 holds, thus X satisfies R(λ, µ; <κ). Suppose that x f | f ∈ I is a sequence of elements of X. Condition (3) in the present theorem furnishes an element x ∈ X, an ultrafilter D over I, and an α ∈ λ such that x is a D-limit point of x f | f ∈ I , and, for every We will show that x ∈ Eᾱ ,β , for every β ∈ µ, where, as in Lemma 2.2, Since x is a D-limit point of x f | f ∈ I , then {f ∈ I | x f ∈ U} ∈ D and this contradicts {f ∈ I | β ∈ f (ᾱ)} ∈ D, since D is a proper filter.
Corollary 2.4. For every topological space X, the following conditions are equivalent.
(2) For every infinite cardinal µ, and for every sequence x f | f ∈ ω ([µ] <ω ) of elements of X, there is a functionally regular ultrafilter D over ω ([µ] <ω ) such that the sequence has some D-limit point in X.
(3) X is Lindelöf, and, for every sequence x f | f ∈ ω ([ω] <ω ) of elements of X, there is a functionally regular ultrafilter D over ω ([ω] <ω ) such that the sequence has some D-limit point in X.
In order to avoid trivial exceptions, when dealing with products, we shall always assume that all topological spaces under consideration are non empty.
Corollary 2.5. Suppose that µ and κ are infinite cardinals, λ is a nonzero cardinal, and let I = λ ([µ] <κ ). Then, for every product X = j∈J X j of topological spaces, the following conditions are equivalent.
(2) For every choice of sequences x f,j | f ∈ I in X j (one sequence for each j ∈ J), there is a functionally regular ultrafilter D over I such that, for every j ∈ J, the sequence x f,j | f ∈ I has some D-limit point in X j (here D is the same for all sequences).
Proof. The equivalence of (1) and (2) is immediate from Theorem 2.3 and the fact that a sequence in a product has a D-limit point if and only if each projection onto each factor has a D-limit point. The equivalence of (1) and (3) is similar to [Sa,Theorem 2.3]. Cf. also [Co]. First, notice that (1) ⇒ (3) is trivial. We shall complete the proof by showing that if (2) fails, then (3) fails. Hence suppose that we can choose sequences x f,j | f ∈ I in X j , for every j ∈ J, such that, for every functionally regular ultrafilter D over I, there is j ∈ J such that the sequence x f,j | f ∈ I has no D-limit point in X j . Pick one such j ∈ J for each functionally regular ultrafilter D over I, and let J ′ ⊆ J be te set of the j's chosen in this way. Since there are ≤ 2 2 |I| functionally regular ultrafilters over I, we have that |J ′ | ≤ 2 2 |I| . Now applying the already proved equivalence of (1) and (2) to Y = j∈J ′ X j we get that Y does not satisfy R(λ, µ; <κ).
It is not clear whether 2 2 |I| is the best possible bound in Condition (3) in Corollary 2.5, in general. As we shall see in Corollary 3.2 below, in the parallel situation in which we allow arbitrary repetitions of factors from a given family, we can indeed get a lower bound. Of course, in certain particular cases, a better bound in Corollary 2.5 can be obtained. For example, it is immediate from Proposition 3.1 below that a product is Menger if and only if all but finitely many factors are compact, and the product of the noncompact factors (if any) is Menger. In particular, a product is Menger if and only if all countable subproducts are Menger. As another example, it follows easily from [L1] that a product is Lindelöf if and only if all subproducts with ≤ ω 1 factors are Lindelöf.
Some of the results from [Sa, Ca] mentioned in the introduction can be obtained as a particular case of Theorem 2.3, by taking λ = 1. In fact, condition R(1, µ; <κ) is nothing but a restatement of [κ, µ]compactness. Moreover, when λ = 1, an ultrafilter D over λ ([µ] <κ ) ∼ = [µ] <κ is functionally regular if and only if it covers µ, that is, {z ∈ [µ] <κ | β ∈ z} ∈ D, for every β ∈ µ. An ultrafilter D over H is (κ, µ)regular if and only if there is a function f : Thus the results in [Ca,Section 3] are the particular case of Theorem 2.3 when λ = 1. See [L2] for a survey on regularity of ultrafilters and applications (notice that [µ] <κ is denoted by S κ (µ) in [L2]).

Product theorems
From Theorem 2.3 and the general theory developed in [L6] about preservation of properties under products, we get that all powers of some space X satisfy R(λ, µ; <κ) if and only if X is D-compact, for some functionally regular ultrafilter over λ ([µ] <κ ). However, in this particular case, a stronger result can be obtained in a direct way.
We first need an easy proposition. As in Definition 2.1, it is sometimes convenient to allow the possibility that µ = ∞. Notice that, in this sense, R(1, ∞; <κ) is what is usually called final κ-compactness, which we shall also call [κ, ∞]-compactness.
If X = α∈λ X α , we shall denote by π α the natural projection onto the α th component.
Proof. By assumption, for every α ∈ λ, there is an open cover W α of X α such that |W α | ≤ µ, and no subset of W α of cardinality < κ is a cover of X α . For every α ∈ λ, let U α be the open cover of X consisting of all the products of the form δ∈λ Y δ , where all Y δ 's equal X δ , except for δ = α, in which case we require that Y δ ∈ W α . The family (U α ) α∈λ is clearly a counterexample to R(λ, µ; <κ). We have to show that, for every choice of subfamilies V α ⊆ U α such that |V α | < κ for every α ∈ λ, we have that α∈λ V α fails to be a cover of X. Indeed, for every α ∈ λ, since no subset of W α of cardinality < κ is a cover of X α , there is x α ∈ X α such that, whenever x ∈ X, and π α (x) = x α , then x belongs to no member of V α . By choosing an x α as above, for each α ∈ λ, and by taking x = (x α ) α∈λ ∈ X, we get that x belongs to no member of α∈λ V α ; thus this last family is not a cover of X. In the statement of the next corollary the expression "product of members of a family F " is always intended in the sense that repetitions are allowed in the product, that is, the same space can occur multiple times.
(4) There is a (κ, µ)-regular ultrafilter D (which can be chosen over [µ] <κ ) such that every member of F is D-compact. (5) All products of members of F are [κ, µ]-compact. (6) All products of members of F satisfy R(λ ′ , µ; <κ), for every nonzero cardinal λ ′ . In particular, every product of members of F satisfies the Menger property for countable covers if and only if so does any product of ≤ 2 2 ω factors, if and only if every product of members of F is countably compact.
(2) ⇒ (3) Let X = j∈J X j be a product of ≤ ν members of F . Since λ ≤ ν, and ν is infinite, then also X λ is (can be reindexed as) a product of ≤ ν members of F . By (2), X λ satisfies R(λ, µ; <κ), and, by Proposition 3.1 with all X α 's equal to X, we get that X is [µ, κ]-compact. We have proved that (3) holds.
The equivalence of (3), (4) and (5) comes from [Ca,Section 3]. It can be also obtained from [L6], and the particular case of Theorem 2.3 when λ = 1, using the remarks at the end of Section 2.
As in [Ca] or [L3,Remark 2.4], there are cases in which the value of ν in Corollary 3.2 can be improved. Just to state a simple example, we can have ν = max{λ, 2 2 µ }, when µ = κ is a regular cardinal or, more generally, when cf µ ≥ κ.
It is trivial to see that if X and Y are topological spaces, f : X → Y is a continuous and surjective function, and X satisfies R(λ, µ; <κ), then Y satisfies R(λ, µ; <κ), too. In particular, if a product satisfies R(λ, µ; <κ), then all subproducts and all factors satisfy it. Thus from Proposition 3.1 we get that if some product X = j∈J X j satisfies R(λ, µ; <κ), then all but at most < λ factors are [κ, µ]-compact. In particular, if a product satisfies the Menger property for countable covers, then all but finitely many factors are countably compact (this extends [A,Proposition 1], where a different terminology is used, and those spaces we call Menger here are called Hurewicz there).
The above remarks can be improved in combination with results from [L1], as we shall show in the next proposition.
By the remarks before the statement of the proposition, it is enough to prove that if Y = γ∈λ Y γ is a product of λ-many spaces, and no Y γ is [κ, µ]-compact, then Y does not satisfy R(λ, µ; κ +n ). So suppose that Y and the Y γ 's are as above. Since a topological space is [κ, µ]-compact if and only if it is both [κ, κ +n ]-compact and [κ +n+1 , µ]-compact, and since λ is infinite, then either |{γ In the latter case we have that Y does not satisfy R(λ, µ; κ +n ), by Proposition 3.1, and recalling that R(λ, µ; κ +n ) is the same as R(λ, µ; < κ +n+1 ).

The Rothberger property
We now compare the Menger and the Rothberger properties, as far as filter convergence is concerned.
Notice that in the proof of Theorem 2.3 we made an essential use of the assumption that κ is infinite. Indeed, for κ = 2, say, for the Rothberger property R(ω, ∞; 1), a similar characterization is not possible, at least, not using ultrafilters. However, some of the arguments in the proof of Theorem 2.3 can be carried over, furnishing a characterization in terms of filters. The results from [L6] do apply also in this more general situation, but the fact that the characterization involves only filters which are not maximal implies that the behavior with respect to products is entirely different. See Corollary 4.2 below.
(2) For every sequence x f | f ∈ I of elements of X, there is a filter F over I such that (a) there isᾱ ∈ λ such that, for every A ∈ F and every β ∈ µ, Proof. If X satisfies R(λ, µ; <κ), and x f | f ∈ I is a sequence of elements of X, then, by Lemma 2.2(3), there areᾱ ∈ λ, and Conversely, if (2) holds, then anyᾱ given by (a) and any point x given by (b) are such that x ∈ β∈µ {x f | f ∈ I, β ∈ f (ᾱ)}, thus Condition (3) in Lemma 2.2 holds.
Of course, Proposition 4.1 holds also when µ and κ are infinite. The main point in Theorem 2.3 is the non trivial result that, in the case when µ and κ are infinite, we can equivalently restrict ourselves to ultrafilters satisfying Condition (2)(a) in Proposition 4.1. Indeed, if some ultrafilter D over λ ([µ] <κ ) satisfies (2)(a), forᾱ ∈ λ, then each Aᾱ ,β belongs to D, since otherwise A = I \ Aᾱ ,β ∈ D, contradicting (2)(a). Thus an ultrafilter D satisfies (2)(a) if and only if it is functionally regular. That is, Condition (2) in Proposition 4.1, when restricted to ultrafilters, becomes Condition (3) in Theorem 2.3.
The notion of F -compactness is usually given only for ultrafilters, since no T 1 space with more than one point is F -compact, when F is a filter which is not maximal [L6]. However, as Proposition 4.1 shows, the notion of an F -limit point has some interest even when F is a filter not maximal. Another example in which general filters are necessary is sequential compactness, dealt with in [L6].
In [L6] we have studied those classes K of topological spaces which can be characterized by means of filter convergence, in the sense that there is a set I and a family P of filters over I such that, for every I-indexed sequences of elements from any space X ∈ K, there is some F ∈ P such that the sequence F -converges to some element of X (here examples of such classes K are furnished by Theorem 2.3 and Proposition 4.1). It has been proved in [L6] that if K allows a characterization as above, and the corresponding family P contains exclusively filters which are not maximal (e. g., as in Proposition 4.1, but not as in Theorem 2.3), then any space X with the property that all powers of X belong to K must be ultraconnected (a topological space is ultraconnected if it does not contain a pair of disjoint nonempty closed sets).
In particular, if all powers of some space X are Rothberger, then X is ultraconnected, and thus satisfies very few separation properties (unless it is a one-element space). In fact, a stronger result follows directly from Proposition 3.1.
In particular, if a product satisfies the Rothberger property (even just for countable covers), then all but a finite number of factors are ultraconnected.
Proof. Immediate from Proposition 3.1, noticing that ultraconnectedness is the same as [2, 2]-compactness (every two-elements open cover has a one-element subcover).
The last statement follows from the remarks before Proposition 3.3.