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Selective screenability, products and topological groups

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##### Selective screenability, products and topological groups

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**III Workshop on Coverings, Selections and Games in**TopologyApril 25-29, 2007Serbia Selective screenability, products and topological groups Liljana BabinkostovaBoise State University**Selection principle Sc(A,B)**A and B are collections of families of subsets of an infinite set. For each sequence (Un: n<∞) of elements of A there is a sequence (Vn: n<∞) such that: 1) Each Vn is a pairwise disjoint family of sets, 2) each Vn refines Un and 3) {Vn: n<∞} is an element of B.**The game Gck(A,B)**The players play a predetermined ordinal number k of innings. In inning n: ONE chooses any On from A, TWO responds with a disjoint refinement Tn of On. TWO wins a play ((Oj,Tj): j< k) if {Tj : j < k } is in B; else ONE wins.**Gck(A,B) and Dimension**O denotes the collection of all open covers of X. For metrizable spaces X, for finite n the following are equivalent: (i) dim (X) = n. (ii) TWO has a winning strategy in Gcn+1(O,O), but not in Gcn(O,O).**Gck(A,B) and Dimension**For metrizable spaces X the following are equivalent: (i) X is countable dimensional. (ii) TWO has a winning strategy in Gc(O,O).**HAVER PROPERTY**For each sequence (εn: n<∞) of positive real numbers 3) {Vn: n<∞}is an open cover of X. there is a corresponding sequence (Vn:n<∞) where 1) each Vn is a pairwise disjoint family of open sets, 2) each element of each Vn is of diameter less than εn, and W.E.Haver, A covering property for metric spaces, (1974)**Property C and Haver property**When A = B = O, the collection of open covers, Sc(O,O) is also known as property C. 1) In metric spaces Sc(O,O) implies the Haver property 2) countable dimension => Sc(O,O) => Sc(T,O) • Notes: • T is the collection of two-element open covers. • Sc(T,O) is Aleksandroff’s weak infinite dimensionality D. Addis and J. Gresham, A class of infinite dimensional spaces I, (1978)**Haver Property does not imply Sc(O,O) Example**X=M L M - complete metric space, totally disconnected strongly infinite dimensional. L – countable dimensional s.t. M L is compact metric space.**Z**Haver property ≠ Property C L Z M L – countable dimensional Z M, Z – zero dimensional (compact subset of totally disconnected space)**Alexandroff ’s Problem**Is countable dimensionality equivalent to weak infinite dimensionality? R.Pol (1981): No. There is a compact metrizable counterexample.**The Hurewicz property**For each sequence (Un: n<∞) of open covers of X Hurewicz Property: there is a sequence (Vn: n<∞) of finite sets such that 1) For each n, Vn Un and 2) each element of X is in all but finitely many of the sets Vn. W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, (1925)**The Haver- and Hurewicz- properties**For X a metric space with the Hurewicz property, the following are equivalent: 1) Sc(O,O) holds. 2) X has the Haver property in some equivalent metric on X. 3) X has the Haver property in all equivalent metrics on X.**Products and Sc(O,O)**(Hattori & Yamada; Rohm,1990) : Let X and Y be topological spaces with Sc(O,O). If X is σ-compact, then XxY has Sc(O,O). (R.Pol,1995):(CH) For each positive integer n there is a separable metric space X such that 1) Xn has Sfin(O,O) and Sc(O,O), and 2) Xn+1 has Sfin(O,O), but not Sc(O,O).**Products and Sc(O,O)**Let X be a metric space which has Sc(O,O) and Xn has the Hurewicz property. Then Xn has propertySc(O,O). If X and Y are metric spaces with Sc(O,O), and if XxY has the Hurewicz property then XxY has Sc(O,O).**Products and the Haver property**Let X be a complete metric space which has the Haver property. Then for every metrizable space Y which has the Haver property, also XxY has the Haver property. Let X and Y be metrizable spaces such that X has the Haver property and Y is countable dimensional. Then XxY has the Haver property. Let X and Y be metrizable spaces with the Haver property. If X has the Hurewicz property then XxY has the Haver property.**Haver and Sc(O,O) in topological groups**Let (G,*) be a topological group and U be an open nbd of the identity element 1G. Open cover of G: Onbd(U)={x*U: xG} Collection of all such open covers of G: Onbd={Onbd(U): U nbd of 1G}**Haver and Sc(Onbd,O) in topological groups**Let (G,*) be a metrizable topological group. The following are equivalent: • G has the Haver property in all left invariant metrics. • G has the property Sc(Onbd, O).**Products and Sc(Onbd,O) in metrizable groups**Let (G,*) be a group which has property Sc(Onbd,O) and the Hurewicz property. Then if (H,*) has Sc(Onbd,O), GxH also has Sc(Onbd,O).**Games and Sc(Onbd,O) in metrizable groups**If (G,*) is a metrizable group then TFAE: • TWO has a winning strategy in Gc(Onbd, O). • G is countable dimensional.**Relation to Rothberger- and Menger-bounded groups**S1(nbd,O) Sc(nbd,O) S1(Onbd,O) Sc(Onbd,O) None of these implications reverse.**New classes of open covers**X-separable metric space CFD: collection of closed, finite dimensional subsets of X FD: collection of all finite dimensional subsets of X**Ocfd and Ofd covers**Ocfd – all open covers U of X such that: X is not in U and for each CCFD there is a UU withC U . Ofd – all open covers U of X such that: X is not in U and for each CFD there is a UU withC U .**Selection principle S1(A,B)**Aand Bare collections of families of subset of an infinite set. For each sequence (Un: n<∞) of elements of A there is a sequence (Vn: n<∞) such that: 1) For each n, Vn Un 2) {Vn: nN } B.**Sc(O,O) and S1(Ofd,O)**Let X be a metrizable space. S1(Ofd,O) => Sc(O,O) Sc(O,O) ≠> S1(Ofd,O)**New classes of weakly infinite dimensional spaces**Sfin(O,O) Sfin(Ocfd,O) Sfin(Ofd,O) S1(Okfd,O) S1(Ocfd,O) S1(Ofd,O) Sc(O,O) Sc(T,O) KCD SCD CD**S1(Ofd,O) and products**If X has property S1(Ofd,O) and Y is countable dimensional, then XxY has property S1(Ofd,O).**S1(Ocfd,O) and products**If X has property S1(Ocfd,O) and Y is strongly countable dimensional, then XxY has property S1(Ocfd,O).**III Workshop on Coverings, Selections and Games in**TopologyApril 25-29, 2007Serbia Thank you!