A characterization of the Menger property by means of ultrafilter convergence

Abstract

Abstract. We characterize various Menger/Rothberger-related properties, 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 2010 Mathematics Subject Classification. Primary 54D20, 54A20; Secondary 54B10, 03E05.

Topological spaces compact with respect to a set of filters

“…This provides a standard method for dealing with the problem, and is what Bernstein [2], Ginsburg and Saks [10], Saks [19] and Caicedo [3], among others, have done in particular cases, as we mentioned in the introduction. We have continued this line of research in [16] for the Menger and the Rothberger properties, here for sequential compactness, and in [17] for [µ, λ]-compactness. See also [18].…”

“…Moreover, these examples show how significant the difference is between the case in which P contains some ultrafilter, and the case in which P contains no ultrafilter (compare the last statement in Theorem 1). Indeed, there are T 1 spaces all whose powers are Menger (they are exactly the compact spaces); on the contrary, in [16,Corollary 4.2] we prove that if some product of T 1 spaces is Rothberger, then all but at most a finite number of the factors are one-element. A somewhat similar situation occurs in the case of sequential compactness, as we are going to discuss in Section 5.…”

“…Actually, we consider even more general properties, which depend on three cardinals. Though Theorem 1 can be applied in this situation, too, in [16,Theorem 2.3] we are able to obtain stronger bounds in a direct way. Anyway, the above characterizations are good examples of the usefulness of allowing non maximal filters in P; indeed, the characterization of the Rothberger property involves a family P consisting of filters none of which is maximal [16, Proposition 4.1, and the comments below].…”

“…In [16] we provide characterizations of the Menger property and of the Rothberger property in terms of sequencewise P-compactness. Actually, we consider even more general properties, which depend on three cardinals.…”

“…Also the Menger, the Rothberger and related properties can be given an equivalent formulation in terms of sequencewise P-compactness. See [16].…”