Open covers and a polarized partition relation

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1 Open covers and a polarized partition relation Marion Scheepers 1 Abstract We show that a strengthened form of a property of Rothberger implies that a certain polarized partition relation holds for the ω covers of a space, and that this partition relation implies a strengthened form of a property of Menger. 2 Let k be a positive integer and let A 1, A 2, B 1,1, B 2,1,...,B 1,k and B 2,k be collections of subsets of an infinite set S. Then the the symbol ( A1 ) A 2 ( ) 1,1 B1,1... B 1,k B 2,1... B 2,k denotes the polarized partition relation, defined as follows: For each A 1 A 1, for each A 2 A 2 and for each f : A 1 A 2 {1,...,k} there are an i {1,...,k} and sets B 1 A 1 and B 2 A 2 such that B 1 B 1,i, B 2 B 2,i, and {f(x, y) :(x, y) B 1 B 2 } = {i}. The polarized partition relation was introduced in Section 9 of [11]; in this paper they also introduced the better known ordinary partition relation (which will not be featured here). The polarized relation is studied extensively in [8] for ordinals. Another class of partition relations known as the square-bracket partition relations was introduced in section 18 of [9]. These three classes of partition relations were considered mostly in connection with cardinal or ordinal numbers, although the ordinary partition relation and the square bracket relation have also been extensively studied in the theory of ultrafilters (see for example [2], [3], [4] and [13]) and the theory of linear order types (see for example [10] and [7]). We discuss here a hybrid of the polarized and of the square bracket relation (hinted at on p. 59 of [6]) in connection with certain open covers of topological spaces. This partition relation is defined as follows: Let A 1, A 2, B 1 and B 2 be collections of subsets of the set S and let k and l be positive integers. Then the symbol A1 B1 A 2 B 2 k/<l denotes the statement that for each A 1 A 1, for each A 2 A 2 and for each f : A 1 A 2 {1,...,k} there are sets B 1 A 1, B 2 A 2, and J {1,...,k} such that B 1 B 1, B 2 B 2, J l and {f(x, y) :(x, y) B 1 B 2 } J. The negation of either of these statements is denoted by replacing the with a. 1 Research supported by NSF grant DMS Keywords and phrases: Rothberger property, Menger property, Polarized partition relation, ω cover. 1

2 From now on let X be fixed a topological space. According to [12] an open cover U of X is an ω cover if X is not an element of U, but for each finite subset of X there is an element of U which contains that finite set. The symbol denotes the collection of (open) ω covers of X. Also according to [12] X is said to be an ɛ space if each ω cover of X has a countable subset which is an ω cover of X. We study the following instance of the weak square bracket polarized partition relation introduced above: (1) k/<l The reason for restricting ourselves to this weaker square-bracket version is as follows: In Theorem 47 of [11] it was shown that for all cardinal numbers ( ) ( ) 1,1 κ κ 1 κ,. This implies that for any infinite space we have κ 1 κ ( ) ( ) 1,1 1 ; just look at an ω cover of minimal cardinality of the 1 space, and apply Theorem 47 of [11]. This indicates that if we are to obtain a positive relation involving ω-covers on both sides of the, we must admit more than one value for the function f. A space has property S 1 (O, O) if there is for each sequence (U n : n = 1, 2, 3,...) of open covers of it a sequence (U n : n =1, 2, 3,...) such that for each nu n U n, and {U n : n =1, 2, 3,...} is a cover of X. This property was introduced by Rothberger in [17], where he used the symbol C to denote this property. A space X has property S 1 (, ) if there is for every sequence (U n : n =1, 2, 3,...)ofω covers of X a sequence (U n : n =1, 2, 3,...) such that for each nu n U n, and {U n : n =1, 2, 3,...} is an ω-cover of X. This notion (but not notation) was introduced by Masaki Sakai wo proved in [18] that for ɛ spaces having property S 1 (, ) is equivalent to having Rothberger s property C in all finite powers. According to Theorems 24 and 25 of [19] and a result of [15] this property is also characterized by the ordinary partition relation for ω covers, and by [20] it is also characterized by the square bracket partition relation for ω covers. Though we have only partial results in this connection, we suspect that also the polarized relation (1) characterizes the property S 1 (, ). We now proceed to give the evidence on which this suspicion is based. Theorem 1 If X has property S 1 (, ), then holds. Proof : Let U and V be ω-covers of X. Since X has property S 1 (, ) we may assume that each of these is countable. Enumerate U bijectively as (U n : n<ω). Let k be a positive integer and let f : U V{1,...,k} be given. 2

3 Choose a chain of subsets of V as follows: Put V 1 = V, and for 1 n<ω choose i n+1 {1,...,k} such that V n+1 = {V V n : f(u n+1,v)=i n+1 }. Since U is an ω cover, fix an i {1,...,k} and n 1 < n 2 <... such that {U n1,u n2,...} and for each j, i nj = i. Apply property S 1 (, ) to the sequence (V n1, V n2,...,v nm,...)ofω-covers of X and select for each m a V m V nm such that {V m : m =1, 2, 3,...} is an -cover of X. We may assume that this enumeration is bijective. At this stage we have that if j m, then f(u nj,v m )=i. Put U 1 = {U n1,u n2,...} and choose for each m a j m+1 {1,...,k} such that U m+1 = {U nr U m : f(u nr,v m+1 )=j m+1 }. Fix a j and m 1 < m 2 <... such that for each rj mr = j, and {V m1,v m2,...} is an ω cover of X. Next we apply S 1 (, ) to the sequence (U m1, U m2,...)ofω-covers of X, as follows: From U mt choose U nht such that for each t, m t <h t <h t+1 and {U nht : t =1, 2, 3,...} is an ω-cover of X. One way of seeing that this can be done is as follows. In [21] it was shown that X has property S 1 (, ) if, and only if, ONE has no winning strategy in the following game, denoted G 1 (, ): In the n th inning ONE chooses an ω cover O n of X; TWO responds by selecting T n O n. Aplay(O 1,T 1,...,O n,t n,...) is won by TWO if {T 1,T 2,T 3,...} is an ω-cover of X; otherwise, ONE wins. Now consider the strategy which calls on ONE to play in the t th inning the set of elements of U mt of the form U nh where h exceeds max{m 1,...,m h } as well as the maximum of all subscripts of elements chosen thus far by TWO. Since this is not a winning strategy, there is a play according to it which is won by TWO. Such a play provides us with a sequence of U nht s as above. But then on {U nh1,u nh2,...} {V m1,v m2,...} the function f takes only the two values, i and j. The space X is said to have property S fin (, ) if there is for every sequence (U n : n =1, 2, 3,...) of (open) ω covers of X a sequence (V n : n =1, 2, 3,...) such that for each n V n is a finite subset of U n, and n=1v n is an ω-cover of X. Combined results of [15] and [19] show that also this property is characterized by a partition relation. This property is intimately connected with another property from classical literature. In [14] Hurewicz showed that a property introduced in [16] by Menger is equivalent to the property S fin (O, O) i.e., for every sequence (U n : n =1, 2, 3,...) of open covers of X there is a sequence (V n : n =1, 2, 3,...) such that each V n is a finite subset of U n, and n=1v n is a cover of X: Property S fin (O, O) is said to be Menger s property. In [15] it was shown that a space has property S fin (, ) if, and only if, all its finite powers have Menger s property. Arkhangel skiǐ showed that having Menger s property in all finite powers is equivalent to the space of real-valued continuous functions with the topology of pointwise convergence having certain convergence properties [1]. 3

4 Theorem 2 If for ɛ space X S fin (, ) ( ) [ ] 1,1, then X has property Proof :Let(U n : n =1, 2, 3,...) be a sequence of ω covers of X. We may assume each U n is countable, and enumerate it as (Um n : m =1, 2, 3,...). Define V to be the nonempty intersections of the form Un U 1 m. n Then V is an ω-cover of X. For each element of V choose a specific representation of the form Un 1 U m n. Define a coloring f : V V{0, 1, 2, 3, 4, 5, 6, 7, 8} such that 0 if n 1 = n 2 and k 1 = k 2 1 if n 1 = n 2 and k 1 <k 2 2 if n 1 = n 2 and k 1 >k 2 3 if n 1 <n 2 and k 1 = k 2 f(un 1 1 U n1 k 1,Un 1 2 U n2 k 2 )= 4 if n 1 <n 2 and k 1 <k 2 5 if n 1 <n 2 and k 1 >k 2 6 if n 1 >n 2 and k 1 = k 2 7 if n 1 >n 2 and k 1 <k 2 8 if n 1 >n 2 and k 1 >k 2 Applying the partition relation to this f we find ω-covers K and L, subsets of V, such that f has no more than two values on K L. One can show that f must have a value in {3, 4, 5} and must have a value in {6, 7, 8}. By considering the cases one sees that value sets {3, 7}, {3, 8}, {4, 6}, {5, 6} or {5, 7} are impossible. Having value set {3, 6}, {4, 7} or {5, 8} gives a sequence (U m : m =1, 2, 3,...) such that for each mu m U m, and {U m : m = 1, 2, 3,...} is an ω-cover of X. The only remaining case is that f has values 4 and 8. An examination of this case leads to a sequence (V n : n =1, 2, 3,...) where for each n V n is a finite subset of U n, and n=1v n is an ω cover of X. Space X satisfies property Split(, ) if there is for each ω cover U of X two ω-covers B 1 and B 2 such that B 1 B 2 = and B 1 B 2 U. This property was introduced in [19] and studied further in [15]. Theorem 3 If for ɛ space X Split(, ) ( ) [ ] 1,1, then X has property Proof : Let U be an ω-cover of X. We may assume that it is countable. Enumerate it bijectively as (U m : m =1, 2, 3,...). Then define f : U U {0, 1, 2} so that 0 if m = n f(u m,u n )= 1 if m<n 2 if m>n. 4

5 Apply the polarized relation to find ω-covers V and W contained in U such that on V W f is two-valued. We show that V and W are disjoint by showing that 0 is not a value of f. Suppose on the contrary that 0 is a value of f on V W. Choose n such that U n V W. Since V and W are ω-covers they are infinite. Thus there are U m Vwith m>n, and U k Wwith k>n. This implies that 1 and 2 are also values of f, so that f takes three distinct values on V Winstead of just two. From now on assume that our spaces are sets of real numbers endowed with the topology inherited from the real line. To clarify the relation among the three theorem above we need to introduce a few cardinal numbers associated with the real line. The minimal cardinality of a cover of the real line by first category sets is denoted cov(m). In [15] it was shown that the minimal cardinality of a set of real numbers which does not have property S 1 (, ) is cov(m). If ω denotes the set of nonnegative integers, then ω ω denotes the set of sequences of nonnegative integers. For f and g such sequences we write f g to denote that lim n (g(n) f(n)) =. Then is a partial ordering of ω ω. A subset D of ω ω is a dominating family if for each f in ω ω there is a g Dsuch that f g. The symbol d denotes the minimal cardinality for a dominating subset of ω ω. In [15] it was shown that d is also the minimal cardinality of a set of real numbers which does not have property S fin (, ). A subcollection U of the collection of subsets of ω is an ultrafilter if for each partition of ω into three disjoint sets, exactly one of these sets belongs to U (though this is not the traditional definition, it is equivalent with the usual one). A family B of subsets of ω is an ultrafilter base if the set {A ω :( B B)(B A)} is an ultrafilter. The minimal cardinality of an ultrafilter base with no finite sets as elements is denoted u. It was shown in [15] that the minimal cardinality of a set of real numbers which does not have property Split(, ) is u. Now let κ be the minimal cardinality of a set of real numbers which does not satisfy the property. By Theorems 1, 2 and 3 we have cov(m) κ min{d, u}. Let ν and δ be uncountable regular cardinal numbers. According to the theorem of [5] it is consistent, relative to the consistency of ZFC, that u = ν and d = δ. Consequently one cannot prove that κ is d, implying that S fin (, ) does not imply the partition relation (1) for l = 3. Similarly, Split(, ) does not imply this relation. It seems that a reasonable first step towards determining if for ɛ spaces S 1 (, ) is characterized by this partition relation would be to answer Problem 1 Is cov(m) =κ? 5

6 References [1] A.V. Arkhangel skiǐ, Hurewicz spaces, analytic sets and fan tightness of function spaces, Soviet Mathematical Doklady 33 (1986), [2] J.E. Baumgartner and A.D. Taylor, Partition theorems and ultrafilters, Transactions of the American Mathematical Society 241 (1978), [3] A. Blass, Ultrafilter mappings and their Dedekind cuts, Transactions of the American Mathematical Society 188 (1974), [4] A. Blass, Amalgamation of nonstandard models of Arithmetic, The Journal of Symbolic Logic 42 (1977), [5] A. Blass and S. Shelah, Ultrafilters with small generating sets, Israel Journal of Mathematics 65 (1989), [6] P. Erdös, A. Hajnal, A. Máté and R. Rado, Combinatorial Set Theory: Partition relations for cardinals, North-Holland Publishing Company (1984). [7] P. Erdös, A. Hajnal and E.C. Milner, Partition relations for η α and for ℵ α saturated models, in Theory of Sets and Topology edited by G. Asser, J. Flachsmeyer and W. Rinow, Berlin (1972), [8] P. Erdös, A. hajnal and E.C. Milner, On set systems having paradoxical covering properties, Acta Mathematica Academiae Scientarum Hungaricae 31 (1978), [9] P. Erdös, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), [10] P. Erdös, E.C. Milner and R. Rado, Partition relations for η α sets, Journal of the London Mathematical Society 3 (1971), [11] P. Erdös and R. Rado, A partition calculus in Set Theory, Bulletin of the American Mathematical Society 62 (1956), [12] J. Gerlits and Zs. Nagy, Some properties of C(X), I, Topology and its Applications 14 (1982), [13] J.M. Henle, A. Kanamori and E.M. Kleinberg, Filters for square -bracket partition relations, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 30 (1984), [14] W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925),

7 [15] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, Combinatorics of open covers (II), submitted. [16] K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik 133 (1924), [17] F. Rothberger, Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae 30 (1938), [18] M. Sakai, Property C and function spaces, Proceedings of the American Mathematical Society 104 (1988), [19] M. Scheepers, Combinatorics of open covers (I): Ramsey Theory, Topology and its Applications 69 (1996), [20] M. Scheepers, Open covers and the square bracket partition relation, Proceedings of the American Mathematical Society, to appear. [21] M. Scheepers, Combinatorics of open covers (III): games, C p (X), submitted. 7

A sequential property of C p (X) and a covering property of Hurewicz.

A sequential property of C p (X) and a covering property of Hurewicz. Marion Scheepers 1 Abstract C p(x) has the monotonic sequence selection property if there is for each f, and for every sequence (σ