SELECTION PRINCIPLES AND INFINITE GAMES ON MULTICOVERED SPACES TARAS BANAKH, LYUBOMYR ZDOMSKYY Contents Introduction 1 1. Multicovered Spaces and their Morphisms 2 2. Classical selection principles 7 3. F-Menger multicovered spaces 11 4. Selection Principles in topological spaces 19 5. Games on Multicovered spaces 26 6. Determinacy of Games related to Selection Principles 30 References 34 Introduction This article is a survey of results included in the forthcoming book [9]. The starting impulse for writing this book came from topological algebra. Trying to ¯nd an inner characterization of subgroups of σ-compact topologi- cal groups, Okunev introduced the concept of an o-bounded group (see [74]): a topological group G is o-bounded if for any sequence (Un)n2! of open neigh- borhoods of theS unit of G there is a sequence (Fn)n2! of ¯nite subsets of G such that G = n2! Fn ¢Un where A¢B = fab : a 2 A; b 2 Bg is the product of two subsets A, B in the group G. The class of o-bounded groups turned out to be much wider than the class of subgroups of σ-compact groups: for example the group R! contains a non-meager dense o-bounded subgroup. To overcome this di±culty M.Tkachenko introduced the narrower class of so-called strictly o-bounded groups, de¯ned with help of the in¯nite game \Open-Finite" played by two players, I and II, on a topological group G: at the n-th inning the ¯rst player selects a neighborhood Un of the unit in G while the second player responds with a ¯nite subset Fn ofSG. At the end of the game the second player is declared the winner if G = n2! Fn ¢ Un. Fol- lowing Tkachenko we de¯ne a topological group G to be strictly o-bounded if the second player has a winning strategy in the game \Open-Finite" on G. The class of strictly o-bounded groups includes all subgroups of σ-compact groups and lies in the class of all o-bounded groups. (Strictly) o-bounded groups were intensively studied last time, see [3], [4], [6], [8], [37], [38], [49], [56], [74], [77], [91]. 1 2 TARAS BANAKH, LYUBOMYR ZDOMSKYY In spirit, o-boundedness is very close to the Menger property introduced by K. Menger [58] in 1924 and studied in detail by W. Hurewicz [40] who in- troduced another property known in topology as the Hurewicz property. Re- call that a topological space X has the Menger property (resp. the Hurewicz property) if for any sequence (un)n2! of open covers of X there is a se- quenceS (vn)n2! such that eachS vn, Tn 2 !, is a ¯nite subcollection of un and X = n2! [vn (resp. X = n2! m¸n [vm ). Our crucial observation is that the o-boundedness of a topological group© (G; ¿) is noting else but theª Menger property applied to the family ¹L = fg ¢ U : g 2 Gg : e 2 U 2 ¿ of open covers of G by left shifts of ¯xed neighborhoods of the unit e of G. This observation naturally led us to the concept of a multicovered space, by which we understand a pair (X; ¹) consisting of a set X and a collection ¹ of covers of X (such a collection ¹ is called a multicover of X). The category of multicovered spaces seems to be the most natural place for considering various concepts such as the Menger and Hurewicz proper- ties of topological spaces or the (strict) o-boundedness of topological groups. Such an abstract approach allows us to prove general results having appli- cations and interpretations in various ¯elds of mathematics as di®erent as Set Theory, General Topology, Descriptive Set Theory, Topological Algebra or Theory of Uniform Spaces. 1. Multicovered Spaces and their Morphisms 1.1. Basic de¯nitions and concepts. By a multicover of a set X we understand any family ¹ of covers of X. A set X endowed with a multicover ¹ will be called a multicovered space (denoted by (X; ¹)) or simply X if the multicover is clear from the context. There are many natural examples of multicovered spaces: ² Each topological space X can be considered as a multicovered space (X; O), where O denotes the family of all open covers of X; ² Every metric space (X; ½) carries a natural multicover ¹½ consisting of covers by "-balls: ¹½ = ffB(x; "): x 2 Xg : " > 0g, where B(x; ") def= fy 2 X : ½(y; x) < "g; ² Every uniform space (X; U) has a multicover ¹U consisting of uni- form covers, i.e. ¹U = ffU(x): x 2 Xg : U 2 Ug, where U(x) = fy 2 X :(x; y) 2 Ug; ² In particular, each Abelian topological group G carries a natural multicover ¹G = ffg + U : g 2 Gg : U 6= ; is open in Gg correspond- ing to the uniform structure of G. Next, we de¯ne some notions related to boundedness in multicovered spaces. A subset B of a set X is de¯ned to be u-bounded with respect to a cover u of X if B ½ [v for some ¯nite subcollection v of u. A subset B of a multicovered space (X; ¹) is de¯ned to be ¹0-bounded, where ¹0 ½ ¹, if B is u-bounded for every u 2 ¹0. If ¹0 = ¹, then we simply say that B is bounded in place of ¹-bounded. The notion of a bounded subset MULTICOVERED SPACES 3 is a natural generalization of such concepts as a precompact subset of a regular topological space (i.e., a subset with compact closure) or a totally bounded subset of a uniform space to the realm of multicovered spaces. A multicovered space X is de¯ned to be σ-bounded if it is a countable union of its bounded subsets. On the family cov(X) of all covers of a set X there is a natural preorder Â: u  v if each u-bounded subset is v-bounded. A multicover space (X; ¹) is de¯ned to be centered if any ¯nite subfamily ± ½ ¹ has an upper bound u 2 ¹ with respect to Á (the latter means that each u-bounded subset of X is ±- bounded). For example, all multicovered spaces described before are centered. We shall say that a multicover ¹ on X is ¯ner than a multicover ´ on X (¹  ´) if for any u 2 ´ there is v 2 ¹ with v  u. Two multicovers ¹; ´ of a set X will be called equivalent if each of them is ¯ner than the other. Now we are ready to introduce three important cardinal characteristics of a multicovered space (X; ¹): ² bÂ(X; ¹), the boundedness character of (X; ¹), is the smallest size j±j of a subcollection ± ½ ¹ such that each ±-bounded subset of X is ¹-bounded; ² cof(X; ¹), the co¯nality of (X; ¹), is the smallest size j´j of a sub- family ´ ½ ¹, equivalent to ¹ (which means that for every u 2 ¹ there exists v 2 ´ with v  u); ² bc(X; ¹), the bounded covering number of (X; ¹), is the smallest size of a cover of X by ¹-bounded subsets; ² bc!(X; ¹) = supfbc(X; ´): ´ ½ ¹ is a countable subcollectiong, the countably-bounded covering number. It is clear that bÂ(X; ¹) · cof(X; ¹) for every multicovered space (X; ¹), and a multicovered space (X; ¹) is σ-bounded i® bc(X; ¹) · !. It is also clear that bc!(X; ¹) · bc(X; ¹) and both the cardinals are equal if bÂ(X; ¹) ·@0. Another important notion related to multicovered spaces is !-boundedness generalizing the !-boundedness of topological groups, see [35]. We shall say that a multicovered space (X; ¹) is !-bounded if every cover u 2 ¹ contains a countable subcover u0. Unfortunately, this does not mean that u0 2 ¹. We de¯ne a multicovered space (X; ¹) to be properly !-bounded if for every cover u 2 ¹ there exists a countable subcover u0 ½ u and a cover v 2 ¹ such that v  u0, i.e. each v-bounded subset is u0-bounded. A multicovered space (X; ¹) is said to be paracompact if for any cover u 2 ¹ there is a cover v 2 ¹ with St(v)  u. Here, as expected, St(v) = fSt(V; v): V 2 vg where St(V; v) = [fU 2 v : U \ V 6= ;g. Proposition 1. Every !-bounded paracompact multicovered space is prop- erly !- bound. 1.2. Morphisms between multicovered spaces. Trying to introduce a proper notion of isomorphic multicovered spaces we came to the conclusion 4 TARAS BANAKH, LYUBOMYR ZDOMSKYY that it is too restrictive to look at morphisms as usual single-valued functi- ons. By a morphism from a multicovered space (X; ¹X ) into a multicovered space (Y; ¹Y ) we shall understand a multifunction © : X ) Y , see [11, section 6] for basic information about multifunctions. Such a morphism © : X ) Y is de¯ned to be: ² uniformly bounded, if for every cover u 2 ¹Y there exists a cover v 2 ¹X such that the image ©(B) of any v-bounded subset B of X is u-bounded; ² perfect, if ©¡1 is uniformly bounded; ² an isomorphic embedding if © is a perfect uniformly bounded mor- phism with Dom(©) = X; ² an isomorphism, if © is a perfect uniformly bounded morphism with Dom(©) = X and Im(©) = Y (equivalently, both © and ©¡1 are isomorphic embeddings). The above three cardinal characteristics of multicovered spaces as well as the (proper) !-boundedness are preserved by isomorphisms between multicov- ered spaces. Moreover, a multicovered space (X; ¹) is properly !-bounded i® it is isomorphic to some multicovered space (Y; º), where º consists of countable covers. Finally we explain the nature of perfect morphisms between multicovered spaces of the form (X; O) where X is a topological space and O is the multicover consisting of all open covers of X.