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Projective Versions of Selection Principles ∗ Maddalena Bonanzinga A, Filippo Cammaroto A, Mikhail Matveev B

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Projective Versions of Selection Principles ∗ Maddalena Bonanzinga A, Filippo Cammaroto A, Mikhail Matveev B View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 157 (2010) 874–893 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Projective versions of selection principles ∗ Maddalena Bonanzinga a, Filippo Cammaroto a, Mikhail Matveev b, a Dipartimento di Matematica, Universitá di Messina, 98166 Messina, Italy b Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA article info abstract Article history: All spaces are assumed to be Tychonoff. A space X is called projectively P (where P is Received 6 June 2008 a topological property) if every continuous second countable image of X is P.Character- Received in revised form 2 December 2009 izations of projectively Menger spaces X in terms of continuous mappings f : X → Rω, of Menger base property with respect to separable pseudometrics and a selection prin- MSC: ciple restricted to countable covers by cozero sets are given. If all finite powers of X primary 54D20 secondary 54A20, 54A25, 54C05, 54C30 are projectively Menger, then all countable subspaces of C p (X) have countable fan tight- ness. The class of projectively Menger spaces contains all Menger spaces as well as all Keywords: σ -pseudocompact spaces, and all spaces of cardinality less than d. Projective versions Lindelöf space of Hurewicz, Rothberger and other selection principles satisfy properties similar to the Menger space properties of projectively Menger spaces, as well as some specific properties. Thus, X is Hurewicz space projectively Hurewicz iff C p (X) has the Monotonic Sequence Selection Property in the Rothberger space sense of Scheepers; β X is Rothberger iff X is pseudocompact and projectively Rothberger. Property (∗) Embeddability of the countable fan space Vω into C p (X) or C p (X, 2) is characterized in Property (γ ) terms of projective properties of X. Projectively Menger space © 2009 Elsevier B.V. All rights reserved. Projectively Hurewicz space Projectively Rothberger space Projectively (∗)-space Projectively (γ )-space Projectively countable space Functionally countable space Pseudocompact space Zero set Cozero set ∗ C -embedded set Zero-dimensional space Cardinal numbers b d p cov(M) add(M) Pseudometric Menger base property Strong measure zero Scattered space σ -space Tightness Fan tightness Strong fan tightness Sequential space * Corresponding author. E-mail addresses: [email protected] (M. Bonanzinga), camfi[email protected] (F. Cammaroto), [email protected], [email protected] (M. Matveev). 0166-8641/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2009.12.004 M. Bonanzinga et al. / Topology and its Applications 157 (2010) 874–893 875 Fréchet space Strictly Fréchet space Weakly Fréchet in the strict sense space Weakly Fréchet space Reznichenko property ω-cover γ -cover Thecountablefanspace Ψ -space C p (X) Monotonic Sequence Selection Property Haver property Property C Arhangelskii’s αi properties QN-space wQN-space AP-space APω-space 1. Introduction With just one exception (Section 8.1), by a space we mean a Tychonoff topological space. We follow the idea of char- acterization of properties of spaces by properties of their continuous images [31,7,11,9]. Let P be a topological property. A.V. Arhangelskii calls XprojectivelyP if every second countable image of X is P. Thus, for example, projective compact- ness is equivalent to pseudocompactness. Arhangelskii considered projective P for P = σ -compact, analytic [9], countable (see [8], under the name ω-simple), and also for Cech-complete,ˇ but in that case the mappings were supposed to be open ([7], see also Tkachuk’s paper [62]), and for π -character, in which case the mapping were supposed to be onto compacta of weight ω1 [10]. In this paper we consider projective versions of the Menger, Hurewicz, Rothberger and some other selection principles. When the work on this paper was already in progress, the authors learned about earlier work of Kocinacˇ [34]. Now we mention the results of Kocinacˇ in many places throughout the paper. U ∈ N Recall that a space X is Menger if for every sequence ( n: n ) of open covers of X, one can select finite subfamilies An ⊂ Un such that {An: n ∈ N} covers X. (In another terminology, the Menger property is called Hurewicz [6]; here we follow the terminology of [55] where the name Hurewicz is used for another property.) Originally, the Menger property was considered for second countable spaces, and several characterizations of this prop- erty were found. The idea of this paper is to extend some conditions equivalent to the Menger property for second countable spaces to all Tychonoff spaces. It turns out that this extension is equivalent to the projective Menger property. Another viewpoint to the projective Menger property is the restriction from all open covers to countable covers by cozero sets (here we have a parallel with pseudocompactness: a space is pseudocompact iff every countable cover by cozero sets contains a finite subcover [15]). This approach is not quite new: the restriction of Menger-style properties to uniform covers was considered in [33], and the restriction to covers of topological groups by translates of a neighborhood of unity in [14] and other papers (see [42] for more references). The paper is organized as follows. In Section 3, first, we consider the conditions equivalent to the projective Menger property. Then we consider some properties of projectively Menger spaces and note that if all finite powers of X are projectively Menger, then all countable subspaces of C p(X) have countable fan tightness. In Sections 4–6 we discuss Hurewicz and other selection principles. Thus, for example, it turns out that X is projectively Hurewicz iff C p (X) has the Monotonic Sequence Selection Property in the sense of Scheepers [52]; β X is Rothberger iff X is pseudocompact and projectively Rothberger. In Sections 7 and 9 we interpret some known facts about αi properties and property AP of C p (X) in terms of projective properties of X. In Section 8 we characterize (non)embeddability of the countable fan space Vω into C p(X) in terms of projective properties of X. 2. Terminology In general, we follow [24] and [8]. I =[0, 1] is the unit interval. 0, 1 : X → R are the functions constantly equal to 0 and 1. For F ⊂ X, χF : X → 2 is the function that equals 1 on F and 0 on X \ F . AfamilyU of subsets of a set X is called an ω-cover of X if X ∈/ U and for every finite F ⊂ X there is U ∈ U such that F ⊂ U ; U is a γ -cover if U is infinite and every x ∈ X is contained in all but finitely many elements of U.Thefamiliesof all open covers, of all open ω-covers and of all open γ -covers are denoted by O, Ω and Γ , respectively. We use cz and cl to denote the restriction to countable covers by cozero sets and by clopen sets, respectively; thus for example Γcz denotes the family of all countable γ -covers by cozero sets. U ∈ F ⊂ U Aspace X is Menger if for every sequence ( n: n ω) of open covers of X onecanpickfinitesubfamilies n n so that {Fn: n ∈ ω}=X. In general, if A and B are two classes of covers, then X satisfies the selection principle Sfin(A, B) 876 M. Bonanzinga et al. / Topology and its Applications 157 (2010) 874–893 if for every sequence (Un: n ∈ ω) of elements of A onecanpickfinitesubfamiliesFn ⊂ Un so that {Fn: n ∈ ω}∈B [55]. O O U ∈ Thus X is Menger if it satisfies Sfin( , ).AspaceX is Hurewicz if for every sequence ( n: n ω) of open covers of X one F ⊂ U ∈ ∈ F can pick finite subfamilies n n so that for every x X, x n for all but finitely many n.AspaceX is Rothberger if for every sequence (Un: n ∈ ω) of open covers of X one can pick Un ∈ Un so that {Un: n ∈ ω}=X. In general, if A and B are two classes of covers, then X satisfies the selection principle S1(A, B) if for every sequence (Un: n ∈ ω) of elements of A one can pick elements Un ∈ Un so that {Un: n ∈ ω}∈B [55]. Thus X is Rothberger if X satisfies S1(O, O). ∗ U ∈ AspaceX has the property ( ) of Gerlits and Nagy [28] if for each sequence ( n: n ω) of open covers of X there is a = { ∈ } ∈ U partition X Xn: n ω such that for each n and m there are k and l such that m < k < l and there are U i i , k i l such that Xn ⊂ {U i : k i l}.AspaceX satisfies the property (γ ) [28] if for every open ω-cover U of X one can pick Un ∈ U so that every x ∈ X is contained in all but finitely many Un.Property(γ ) is equivalent [27] to the following formally stronger property (γ1): for every sequence of (Un: n ∈ N) of open ω-covers of X one can pick Un ∈ Un so that every x ∈ X is contained in all but finitely many Un.Property(γ ) implies property (∗) [28]. X is called an -space if all finite powers of X are Lindelöf [28]. A space X is scattered if for every nonempty A ⊂ X there is a ∈ A which is isolated in A. X is called a σ -space if every Fσ -set in X is a Gδ-set [37]. AsubsetF ⊂ ωω is bounded if there is a g ∈ ωω such that for every f ∈ F , f (n) g(n) for all but finitely many n.A subset G ⊂ ωω is dominating if for every f ∈ ωω there is a g ∈ G such that f (n) g(n) for all but finitely many n.
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