Selective covering properties of product spaces

Abstract

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz $\sone(\Ga,\Op)$ space satisfies $\sone(\Ga,\Op)$. On the other hand, assuming \CH{}, for each Sierpi\'nski set $S$ there is a Luzin set $L$ such that $L\x S$ can be mapped onto the real line by a Borel function. \item Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. \item Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits--Nagy. \item Assuming $\fd=\aleph_1$, every productively Lindel\"of space is productively Hurewicz, productively Menger, and productively Scheepers. \ee A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than $\add(\cN)$, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy with $\add(\cN)<\cov(\cM)$. Our results improve upon and unify a number of results, established earlier by many authors.Comment: Submitted for publicatio

Products of general Menger spaces

“…1 and 1.2] proved the same result using weaker hypotheses). It was later shown that, in the realm of hereditarily Lindelöf spaces, every productively Lindelöf space is productively Hurewicz and productively Menger [16,Theorem 8.2]. In our previous paper, we proved that in that realm, every productively Menger space is productively Hurewicz [19,Theorem 4.8].…”

“…(And if not, does (Hurewicz, Lindelöf ) × ⊆ Hurewicz × ? A scale is a ≤ * -increasing sequence { s α : α < d } ⊆ [N] ∞ that is ≤ ∞ -unbounded in [N] ∞ .If S is a scale then, in the realm of hereditarily Lindelöf spaces, the real space S ∪ [N] <∞ is productively Hurewicz and productively Menger[16, Theorem 6.2]. A positive solution of the following problem implies the same assertion for general spaces.…”

“…We identify the Cantor space {0, 1} N with the family P(N) of all subsets of the set N. We denote by [N] <∞ the family of finite subsets of N, so that P(N) = [N] ∞ ∪ [N] <∞ . Since we identify every set a ∈ [N] ∞ with its increasing enumeration, we have [N] The following observation can be proved using earlier methods [16,Theorem 7.3]. Proposition 1.2.…”

Products of Menger spaces in the Miller model

“…As the following fact established in [21] shows, the inequality u < g imposes strong restrictions on the structure of Menger spaces. Lemma 2.3.…”

A coanalytic Menger group that is not $\sigma$-compact

“…A space X is called productively P if X × Y has the property P for each space Y satisfying P . Productively P properties have been studied by many authors (see, e.g., [3,27,34]). …”