SELECTIVE COVERING PROPERTIES OF PRODUCT SPACES, II: γ SPACES ARNOLD W. MILLER, BOAZ TSABAN, AND LYUBOMYR ZDOMSKYY Abstract. We study productive properties of γ spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: (1) Solving a problem of F. Jordan, we show that for every unbounded tower set X ⊆ R of cardinality ℵ1, the space Cp(X) is productively Fr´echet–Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova– Scheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers–Tall, we show that the properties γ and Gerlits–Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel’ski˘ı duality to obtain results concerning local properties of function spaces and countable topo- logical groups. 1. Introduction For a Tychonoff space X, let Cp(X) be the space of continuous real-valued functions on X, endowed with the topology of pointwise convergence, that is, the topology inherited from the Tychonoff product RX . In their seminal paper [14], Gerlits and Nagy characterized the property that the space Cp(X) is Fr´echet–Urysohn—that every point in the closure of a set arXiv:1310.8622v4 [math.LO] 10 Oct 2018 is the limit of a sequence of elements from that set—in terms of a covering property of the domain space X. We study the behavior of this covering property under taking products with spaces possessing related covering properties. By space we mean an infinite topological space. Whenever the space Cp(X) is considered, we tacitly restrict our scope to Tychonoff spaces. The concrete examples constructed in this paper are all subsets of the real line. 1991 Mathematics Subject Classification. Primary: 26A03, Secondary: 03E75, 03E17. Key words and phrases. Gerlits–Nagy property γ, Gerlits–Nagy property (*), Menger property, Hurewicz property, Rothberger property, Sierpi´nski set, selection principles, Cohen forcing, special sets of real numbers, Cp theory, selectively separable, countable fan tightness. 1 2 A.MILLER,B.TSABAN,ANDL.ZDOMSKYY The covering property introduced by Gerlits and Nagy is best viewed in terms of its relation to other, selective covering properties. The framework of selection principles was introduced by Scheepers in [29] to study, in a uniform manner, a variety of properties introduced in several mathematical contexts, since the early 1920’s. Detailed introductions are available in [19, 34, 43, 28]. We provide here a brief one, adapted from [26]. Let X be a space. We say that U is a cover of X if X = U, but X is not covered by any single member of U. Let O(X) be the family of all openS covers of X. When X is considered as a subspace of a larger space Y , the family O(X) consists of the covers of X by open subsets of Y . Define the following subfamilies of O(X): U ∈ Ω(X) if each finite subset of X is contained in some member of U. U ∈ Γ(X) if U is infinite, and each element of X is contained in all but finitely many members of U. Some of the following statements may hold for families A and B of covers of X. A ( B ): Each member of A contains a member of B. S1(A , B): For each sequence h Un ∈ A : n ∈ N i, there is a selection h Un ∈ Un : n ∈ N i such that {Un : n ∈ N} ∈ B. Sfin(A , B): For each sequence h Un ∈ A : n ∈ N i, there is a selection of finite sets N B hFn ⊆ Un : n ∈ i such that n Fn ∈ . U A B A N fin( , ): For each sequence hS Un ∈ : n ∈ i, where no Un contains a finite sub- cover, there is a selection of finite sets hFn ⊆ Un : n ∈ N i such that { Fn : n ∈ N} ∈ B. S We say, e.g., that X satisfies S1(O, O) if the statement S1(O(X), O(X)) holds. This way, the notation S1(O, O) stands for a property (or a class) of spaces. An analogous conven- tion is followed for all other selection principles and families of covers. Each nontrivial property among these properties, where A and B range over O, Ω and Γ, is equivalent to one in Figure 1 [29, 18]. Some of the equivalences request that the space be Lindel¨of. All spaces constructed in this paper to satisfy properties in the Scheepers Diagram are Lindel¨of. Moreover, they are all subspaces of R. In the Scheepers Diagram, an arrow denotes implication. We indicate below each class P its critical cardinality non(P ), the minimal cardinality of a space not in the class. These cardinals are all combinatorial cardinal characteristics of the continuum, details about which are available in [8]. Following the convention in the field of selection principles, influenced by the monograph [5], we deviate from the notation in [8] by denoting the family of meager (Baire first category) sets in R by M. The properties Ufin(O, Γ), Sfin(O, O) and S1(O, O) were first studied by Hurewicz, Menger and Rothberger, respectively. γ spaces were introduced by Gerlits and Nagy [14] as the Ω spaces satisfying ( Γ ). Gerlits and Nagy proved that, for a space X, the space Cp(X) is Fr´echet–Urysohn if and only if X is a γ space. PRODUCTS OF γ SPACES 3 Ufin(O, Γ) / Ufin(O, Ω) / Sfin(O, O) ♦ ♥ ♦♦ 7 b ♥♥♥ 7 d ✉ d ♦♦♦ ♥♥♥ ✉✉ : ♦♦ ♥ ✉✉ ♦♦♦ ✉✉ ♦♦ Sfin(Γ, Ω) ✉✉ ♦♦♦ ✉✉ ♦♦ ♦♦ d ✉✉ ♦♦ ♦♦ 7 ✉✉ ♦♦♦ ♦♦♦ O ✉✉ S1(Γ, Γ) / S1(Γ, Ω) / S1(Γ, O) b d d O O O Sfin(Ω, Ω) ♣♣8 d ♣♣♣ ♣♣♣ Ω S S Γ / 1(Ω, Ω) / 1(O, O) p cov(M) cov(M) Figure 1. The Scheepers Diagram We also consider the classes of covers B, BΩ and BΓ, defined as O, Ω and Γ were defined, replacing open cover by countable Borel cover. The properties thus obtained have rich history of their own [38], and the Borel variants of the studied properties are strictly stronger than the open ones [38]. Many additional—classic and new—properties are studied in relation to the Scheepers Diagram. Definition 1.1. Let P be a property (or class) of spaces. A space X is productively P if X × Y has the property P for each space Y satisfying P . In Section 2 we construct productively γ spaces in R from a weak hypothesis. In Section 3 we construct, using the Continuum Hypothesis, two γ spaces in R whose product is not Menger. In Section 4 we use our results to solve a large number of problems, from the literature and from the folklore of selection principles. In Section 5 we determine the effect of Cohen forcing on γ spaces, Hurewicz spaces, and Gerlits–Nagy (*) spaces. In Section 6 we use our results together with Cp theory, to obtain new results concerning local and density properties of function spaces. In the last section, we prove that every product of an unbounded tower set and a Sierpi´nski set satisfies S1(Γ, Γ). 2. Productively γ spaces in R Recall the Gerlits–Nagy Theorem that a space X is a γ space if and only if the space Cp(X) is Fr´echet–Urysohn. In his papers [16, 17], F. Jordan studied the property that Cp(X) is productively Fr´echet–Urysohn. (In this case, it is said in [16, 17] that the space X is a productive γ-space. Since this terminology is admitted in [17] to be confusing, we avoid 4 A.MILLER,B.TSABAN,ANDL.ZDOMSKYY it here.) We begin with a short proof of a result of Jordan. In the proof, and later on, we use the following observations. Lemma 2.1. Let P be a class of spaces that is hereditary for closed subsets and is preserved by finite powers. Then for all spaces X and Y such that the disjoint union space X ⊔ Y satisfies P , the product space X ×Y satisfies P , too. In particular, if P is preserved by finite unions, then it is preserved by finite products. Proof. We prove the first assertion. As P is preserved by finite powers, the space (X ⊔ Y )2 satisfies P . As X × Y is a closed subset of (X ⊔ Y )2, it satisfies P , too. If the disjoint union space X ⊔ Y is a γ space, then so is the product space X × Y [23, Proposition 2.3]. Corollary 2.2. Let X and Y be spaces. The disjoint union space X ⊔ Y is a γ space if and only if the product space X × Y is. The following observation is made in [16, Corollary 24]. Proposition 2.3 (Jordan). Let X be a space. If the space Cp(X) is productively Fr´echet– Urysohn, then the space X is productively γ. Proof. Let Y be a γ space. To prove that X × Y is a γ space, we may assume that the spaces X and Y are disjoint. By the Gerlits–Nagy Theorem, the space Cp(Y ) is Fr´echet– Urysohn. Thus, the space Cp(X ⊔ Y ) = Cp(X) × Cp(Y ) is Fr´echet–Urysohn. Applying the Gerlits–Nagy Theorem again, we have that X ⊔ Y is a γ space. Apply Corollary 2.2. Some of the major results concerning the property that Cp(X) is productively Fr´echet– Urysohn are collected in the following theorem.