SELECTIVE VERSIONS OF CHAIN CONDITION-TYPE PROPERTIES 5 1 0 LEANDRO F. AURICHI, SANTI SPADARO, AND LYUBOMYR ZDOMSKYY 2 y a Abstract. We study selective and game-theoretic versions of properties like the M ccc, weak Lindelo¨fness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal in- 3 variants of the continuum. 1 ] N G 1. Introduction . h t a Chain conditions provide a measure of how small is a space, from a topological m point of view. For example, a space has the countable chain condition (ccc) if it does [ not contain an uncountable family of pairwise disjoint non-empty open sets. This is 2 one of the weakest chain conditions one can put on a space, while separability may v be considered one of the strongest. Of course, every separable space satisfies the ccc. 7 7 Todorcevic [15] surveys a wealth of chain conditions that are between the ccc and 1 separability, elaborating on their classifying power, in the sense that the better the 0 0 space, the greater number of chain conditions it identifies. A few examples of this . 2 phenomenon are Knaster’s result that separability and Knaster’s property (that is, 0 every uncountable family of open sets contains an uncountable family where each 5 pair of elements meets) are equivalent on ordered continua and Shapirovskii’s result 1 : that separability and the Shanin condition (that is, point-countable families of open v i sets are countable) are equivalent for compact spaces of countable tightness. These X are all ZFC results, but the picture is even clearer in certain models of set theory: r a for example, the ccc and separability are equivalent for linearly ordered spaces under MA . Some of these characterizations even offer topological equivalents of certain ω1 combinatorial principles. Let us just mention Todorcevic and Velickovic’s result that MA is equivalent to the statement that every compact first-countable ccc space is ω1 separable. In more recent times the framework of selection principles in mathematics and topological games has been applied to chain conditions offering more examples of the 2000 Mathematics Subject Classification. Primary: 54A25, 03E17, 91A44; Secondary: 54D35, 54D10. Key words and phrases. chainconditions,selectivelyccc,selectionprinciples,weaklyRothberger, topological games, cardinal inequality. 1 2 LEANDROF. AURICHI,SANTI SPADARO,ANDLYUBOMYRZDOMSKYY phenomenon underlined by Todorcevic. A typical selective chain condition is the one considered by the first-named author in [1] which states that one can diagonalize a familywithadenseunionfromacountablesequenceofmaximalpairwisedisjointfam- ilies of non-empty open sets. Daniels, Kunen and Zhou introduced a game-theoretic strengthening of this property in [9] by using a two-player game where each player plays an inning per natural number and at a given inning, the first player picks a maximal pairwise disjoint open family, while the second player picks an open set from it. The second player wins if the set of all open sets he picked has dense union. If the second player has a winning strategy in this game on a given space X, then X is selectively ccc, which in turn implies that X is ccc. Unfortunately, there are countable spaces failing the selective versions of the ccc (see [1]), so separability alone does not appear to play any role in this context. What should take the role of separability when dealing with selective chain conditions is countable π-weight. Recall that a π-base for a space X is a family P of non-empty open sets such that for every non-empty open U ⊂ X there is P ∈ P such that P ⊂ U. The π-weight of a space is then the minimum cardinality of a π-base. It is easy to see that in every space with a countable π-base the second player has a winning strategy in the ccc game. The second-named author proved in [7] that this is actually equivalent to having a countable π-base for spaces with a countable local π-base at every point. A selective version and a game-theoretic version of separability were defined by Scheepers in [13] in a similar way. The former turns out to be equivalent to a countable π-base for compact spaces and the latter is equivalent to a countable π-base for all regular spaces. These results seem to suggest that the same classifying ability of traditional chain condition is shared by their selective versions, as long as one takes countable π-weight as the ultimate selective chain condition. Even though they measure the topological smallness of a space, chain conditions seldom put any bound on its cardinality. Indeed, while a separable regular space can- c nothave cardinality largerthan2 , therearecccspaces ofarbitrarilylargecardinality: it suffices to consider the Cantor Cube 2κ, where κ is any cardinal. Nonetheless, chain conditions featureprominently inafewclassical cardinal inequalities. Aresult ofHaj- nalandJuhászstatesthatthecardinalityofacccfirstcountablespacedoesnotexceed the continuum. An interesting partial generalization of this theorem, that also gen- eralizes Arhangel’skii’s theorem on the cardinality of first-countable compacta, is due to Bell, Ginsburg and Woods. They proved that the cardinality of a first-countable weakly Lindelo¨f normal space does not exceed the continuum. Weakly Lindelo¨f means that every open cover has a countable subcollection with dense union, a condition which is easily seen to be satisfied by all ccc spaces. The question of whether normal- ity can be relaxed to regularity in this result has remained open, but Angelo Bella gave a partial answer to it in [7] by considering the natural game-theoretic strengthening of the weak Lindelo¨f property. In the third section of our paper we prove this SELECTIVE VERSIONS OF CHAIN CONDITION-TYPE PROPERTIES 3 game is equivalent to a sort of dual of Berner and Juhász’s classical point-picking game and exploit this equivalence to give a short proof to Bella’s Theorem. Anotherreasonforourinterest ingame-theoreticstrengthenings ofchainconditions is that they provide an unexpected ZFC partial positive answer to the old problem of the productivity of the ccc. It was already known by Kurepa that the square of a Suslin Line is not ccc. Thus, consistently, the countable chain condition is not productive. Both Knaster’s property and the Shanin condition are productive, and one can use these results along with the above mentioned equivalences to prove that under MA the ccc is productive. In [9] Daniels, Kunen and Zhou proved in ZFC ω1 that if player II has a winning strategy in the ccc game on every factor of an arbitrary product then it also has it on the full product. In section 2 we deal with selective properties. In particular, we characterize the selective ccc on Pixley-Roy hyperspaces and we give a consistent topological characterization of cov(M) by means of the weak Rothberger property. In section 3 we concentrate on game versions. We prove preservation results in finite unions and products and give a characterization of the weak Rothberger game that we then exploit to prove cardinal inequalities in topological spaces. We then construct counterexamples proving the sharpness of our inequalities. Given a space X with topology τ, we fix some notation about families of open covers and families of subsets of X. • O = {U : U ⊂ τ ∧ U = X}. • O = {U : U ⊂ τ ∧ U = X}. D S • Ω = {U : U ⊂ τ ∧(∀F ∈ [X]<ω∃O ∈ U : F ⊂ O}. S • D = {D ⊂ X : D = X}. • D = {O ∈ τ : O = X}. O An element of Ω is usually known as an ω-cover of X. Let’s recall some basic selection principles and two-person infinite games we will deal with in our paper. Let A,B ⊂ P(X). • We say that X satisfies Sκ(A,B) if for every sequence {A : α < κ} ⊂ A, we 1 α can choose B ∈ A such that {B : α < κ} ∈ B. α α α • We say that X satisfies Sκ (A,B) if for every sequence {A : α < κ} ⊂ A, fin α we can choose B ∈ [A ]<ω such that {B : α < κ} ∈ B. α α α • We denote by Gκ(A,B) the two-person game in κ many innings such that, at 1 S inning α < κ, player one picks A ∈ A and player two picks B ∈ A . Player α α α two wins if {B : α < κ} ∈ B. α • We denote by Gκ (A,B) the two-person game in κ many innings such that, fin at inning α < κ, player one picks A ∈ A and player two picks B ∈ [A ]<ω. α α α Player two wins if {B : α < κ} ∈ B. α S 4 LEANDROF. AURICHI,SANTI SPADARO,ANDLYUBOMYRZDOMSKYY The properties Sω(O,O) and Sω (O,O) are now known as Rothberger and Menger 1 fin respectively. Moreover, we recall that X satisfies Sω(Ω,Ω) if and only if every finite 1 power of X is Rothberger. 2. Selective versions In [13], Marion Scheepers defined a natural selective version of separability which is now known as R-separability (see [4]). The R in the name comes, of course, from Rothberger. Definition 2.1. A space is R-separable if it satisfies Sω(D,D). In other words, for 1 every sequence {D : n < ω} of dense sets there is a point x ∈ D , for every n < ω n n n such that {x : n < ω} is dense in X. n R-separabilityismuchstrongerthanseparability. Forexample, itimpliesthatevery dense set is separable, and by a result of Juhász and Shelah [10], this is equivalent to countable π-weight in the realm of compact spaces. Moreover, there are even examples of countable spaces that are not R-separable (see [5] and [14]). In [1] the first named author introduced the following weakening of R-separability, following a suggestion of Sakai. Definition 2.2. We say that the space X has property S if it satisfies Sω(D ,D), 1 O that is, for every sequence {O : n < ω} of open dense sets we can pick points x ∈ O n n n such that {x : n < ω} is dense. n Oneof the most interesting features about property S is that it lies strictly between R-separability and a natural selective version of the countable chain condition that was introduced by Scheepers in [12]. Definition 2.3. We say that X is selectively ccc if it satisfies Sω(O ,O ). In other 1 D D words, for every sequence {U : n < ω} of open families, such that U is dense for n n every n < ω we can pick an open set U ∈ U such that {U : n < ω} is dense. n n n S S Obviously, every space having property S is selectively ccc, but the converse does not hold, since 2κ is selectively ccc for every cardinal κ (see [1]), but fails to have property S for κ > c because it’s not even separable then. Let X be a space and set PR(X) = [X]<ω. There is a natural topology on PR(X) called the Pixley-Roy topology. A basic open neighbourhood of F ∈ PR(X) is a set of the form [F,U] = {G ∈ PR(X) : F ⊂ G ⊂ U}, where F ⊂ U and U ⊂ X is open in the topology on X. While other selective properties have been characterized on PR(X) (see, for example, [11] and [8]), the problem of characterizing the selective ccc of PR(X) has SELECTIVE VERSIONS OF CHAIN CONDITION-TYPE PROPERTIES 5 remained open. We introduce a new selection principle to provide such a characterization and then reduce the selective ccc of the Pixley-Roy hyperspace of a separable metrizable space X to a well-known selective covering property of X. Definition 2.4. Let (X,τ) be a topological space, and F ⊂ {(F,U) ∈ [X]<ω × τ : F ⊂ U}. We call F an ω-double cover if for every pair (G,V) ∈ [X]<ω×τ such that G ⊂ V there is (F,U) ∈ F such that F ⊂ V and G ⊂ U. The family of all ω-double covers will be indicated with Ω2. Proposition 2.5. Sω(Ω2,Ω2) ⇒ Sω(Ω,Ω). 1 1 Proof. Just note that if U is an ω-cover then {(∅,U) : U ∈ U } is an ω-double n n (cid:3) cover. Theorem 2.6. PR(X) has Sω(O ,O ) if and only if X has Sω(Ω2,Ω2). 1 D D 1 Proof. For the direct implication, let {F : n < ω} ⊂ Ω2. Now let O = {[F,U] : n n (F,U) ∈ F }. Then {O : n < ω} ⊂ O (PR(X)). By the Sω(O ,O ) property of n n D 1 D D PR(X) we can find [F ,U ] ∈ O such that {[F ,U ] : n < ω} ⊂ O . We then have n n n n n D that {(F ,U ) : n < ω} ∈ Ω2. Indeed, let (F,U) ∈ {(F,U) ∈ [X]<ω ×τ : F ⊂ U}. n n Then [F,U]∩[F ,U ] 6= ∅ for some n. So there is H extending both F and F such n n n that H ⊂ U and H ⊂ U, and that implies F ⊂ U and F ⊂ U. n n n Vice versa, suppose X has Sω(Ω2,Ω2) and let {U : n < ω} be a sequence of open 1 n families with dense union in PR(X). Without loss of generality we can assume that U ismadeupofbasicopensets. LetnowF = {(F,U) : [F,U] ∈ U }. ThenF ∈ Ω2, n n n n for every n < ω. Hence we can find (F ,U ) ∈ F such that {(F ,U ) : n < ω} ∈ Ω2. n n n n n Now by the same argument {[F ,U ] : n < ω} has dense union in PR(X). (cid:3) n n Corollary 2.7. If PR(X) satisfies Sω(O ,O ) (that is, PR(X) is selectively ccc) 1 D D then X satisfies Sω(Ω,Ω). 1 Corollary 2.8. If X satisfies Sω(Ω2,Ω2) and U ⊂ X is open then U satisfies 1 Sω(Ω2,Ω2). 1 Proof. If X satisfies Sω(Ω2,Ω2) then PR(X) is selectively ccc. Now PR(U) = [∅,U] 1 is an open subspace of a selectively ccc space and hence it’s selectively ccc. But then U satisfies Sω(Ω2,Ω2). (cid:3) 1 Theorem 2.9. Let X be a second-countable space. Then X is Sω(Ω2,Ω2) if and only 1 if X is Sω(Ω,Ω). 1 Proof. ThedirectimplicationisclearbyProposition2.5. Fortheconverseimplication, let {B : n < ω} be a countable base for X which is closed under finite unions. Let n {I : k < ω} be a partition of ω into infinite sets. Let {U : n < ω} ⊂ Ω2. We can k n assume without loss of generality that for every n < ω and for every (F,U) ∈ U n there is k < ω such that U ⊂ B . Moreover, we can assume that for every n < ω, k 6 LEANDROF. AURICHI,SANTI SPADARO,ANDLYUBOMYRZDOMSKYY and for every (F,U) ∈ U , if V is an open subset of X such that F ⊂ V ⊂ U then n (F,V) ∈ U . For every n ∈ I , define families V as follows: n k n V = {U : (F,U) ∈ U ∧U ⊂ B } n n k Then V is an ω-cover of B and hence we can pick an element U ∈ V for every n k n n n ∈ I such that {U : n ∈ I } is an ω-cover of B . Let F ∈ [X]<ω be such that k n k k n F ⊂ U and (F ,U ) ∈ U . We claim that V = {(F ,U ) : n < ω} is an ω-double n n n n n n n cover for X. Indeed, let (G,V) be any pair, where G ∈ [X]<ω, V ⊂ X is open and G ⊂ V. Let k < ω be such that G ⊂ B ⊂ V. Then we can find a n ∈ I such that k k G ⊂ U . Now F ⊂ U ⊂ B ⊂ V and hence V is actually an ω-double cover for X. n n n k (cid:3) Corollary 2.10. Let X be a separable metrizable space. Then PR(X) is selectively ccc if and only if every finite power of X is Rothberger. The weak Lindelo¨f property is a covering property that may be considered a chain condition, since it is a consequence of the ccc. We finish this section by considering the natural question of when a weakly Lindelo¨f space satisfies the selective version of weak Lindelo¨fness, that is, Sω(O,O ). As in [2], we will call this property the weak 1 D Rothberger property. Theorem 2.11. Let X be a weakly Lindelöf space such that πw(X) < cov(M). Then X is weakly Rothberger. Proof. Recall that the space ωω with its usual topology is homeomorphic to the irrationals and cov(M) can be characterized as the least cardinal of a cover of the irrationals by nowhere dense sets. Let{B : α < κ}enumerateaπ-baseofX, forsomeκ < cov(M). Let{U : n < ω} α n be a sequence of open covers. Since X is weakly Lindelo¨f we can find a countable subcollection {Un : k < ω} of U such that Un is dense. Define N to be the k n k<ω k α following set: S N = {f ∈ ωω : (∀n)(B ∩Un = ∅)} α α f(n) Claim: The set N is nowhere dense in ωω for every α < κ. α Proof of Claim. It will suffice to prove that for every α < κ, the set ωω\N is dense in α ωω. Let [σ] bea basic open set in ωω, where σ ∈ ωn and [σ] = {f ∈ ωω : f ⊃ σ}. n<ω Let k = dom(σ) and pick j < ω such that B ∩ Uk 6= ∅. Let τ = σ ∪ {(k,j)} and α S j note that [τ] ⊂ [σ] and any function in the open set [τ] misses N . △ α Since κ < cov(M) we can pick f ∈/ N . Then {Un : n < ω} is the selection α<κ α f(n) showing that X is weakly Rothberger. (cid:3) S SELECTIVE VERSIONS OF CHAIN CONDITION-TYPE PROPERTIES 7 Theorem 2.12. Under cov(M) < b there is a compact space which is not weakly Rothberger and has π-weight cov(M). Proof. Assume that cov(M) < b. We claim that β(cov(M)) is the required example. We say that F ⊂ ωω satisfies property (P) if for every g ∈ ωω there is f ∈ F such that f(n) 6= g(n) for every n < ω. Claim. There is a family F ⊂ ωω of cardinality cov(M) satisfying property (P) such that f(n) < b(n) for some b ∈ ωω and every f ∈ F. Proof of Claim. ByLemma2.4.2of[3]thereisafamilyF′ ⊂ ωω ofcardinalitycov(M) satisfying (P). Let {h : α < cov(M)} be an enumeration of F′. α Since cov(M) < b, we can fix a function b ∈ ωω such that h <∗ b, for every α α < cov(M). Now, for every α < cov(M) there is n < ω such that h (n) < b(n) for α α every n ≥ n . For every α < cov(M), n < ω and i < b(n) define: α i if n < n hi (n) = α α h (n) if n ≥ n ( α α It’seasytoseethatF = {hi : i < b(n),α < cov(M)}isasubfamilyofωω satisfying α (P) and such that h(n) < b(n), for every h ∈ F. △ Let {f : α < cov(M)} be an enumeration of F and for every n let us consider the α following finiteclopencover ofβ(cov(M)): U = {Un : k < b(n)}, where Un = β({ξ < n k k cov(M) : f (n) = k}) and β(A) is the set of all ultrafilters on cov(M) containing ξ A. We claim that the sequence {U : n ∈ ω} witnesses that β(cov(M)) fails to have n Sω(O,O ). Indeed, suppose that Un is dense in β(cov(M)) for some g ∈ ωω. 1 D n∈ω g(n) Then, in particular, cov(M) ⊂ Un . nS∈ω g(n) But since F satisfies property (P), there exists ξ < cov(M) such that f (n) 6= g(n) S ξ for all n, which means that ξ 6∈ Un for all n, and that is a contradiction. (cid:3) g(n) Corollary 2.13. Assume cov(M) < b. Then cov(M) can be characterized as the minimum π-weight of a non-weakly Rothberger weakly Lindelöf space. Question 2.14. Is it true in ZFC that cov(M) is the minimum π-weight of a non- weakly Rothberger weakly Lindelöf space? 3. Game versions The property S defined in the previous section has a natural game version. Definition 3.1. We say that X has property S+ if the second player has a winning strategy in the game Gω(D ,D). This is the two person game of countable length 1 O where at inning n player one picks a dense open set O and player two picks a point n x ∈ O . Player two wins if the set of all points he picked is dense in X. n n 8 LEANDROF. AURICHI,SANTI SPADARO,ANDLYUBOMYRZDOMSKYY Lemma 3.2. Property S+ is hereditary for open sets. Proof. Let X be a space with property S+ and U ⊂ X be a non-empty open subset. Let τ be a winning strategy for player II in Gω(D ,D) on X. Let σ be the strategy 1 O assigning to the open dense subset O of U the point τ(O ∪Int(X \W)) if this last point is in O, and any point of O otherwise. Then σ is a winning strategy for player II in Gω(D ,D) on O. (cid:3) 1 O The following fact is also clear. Lemma 3.3. If D is dense in X and D has property S+ then X also has property S+. Proposition 3.4. Property S+ is preserved by finite unions. Proof. Once this is proved for the union of two spaces, the result will follow easily by induction, so let X be a topological space and A and B be subspaces with property S+ such that X = A∪B. If Int(A)∩A = ∅ then B is dense in X and we are done. Similarly, if Int(B) ∩ B = ∅ we are done. So we can assume that Int(A) ∩ A and Int(B) ∩B are both non-empty subsets of X. By Lemma 3.2 we can fix a winning strategy σ for player two on Int(A) ∩A and a winning strategy σ for player two A B on Int(B) ∩ B in the game Gω(D ,D). The set Int(A) ∪ Int(B) is a dense open 1 O subset of X. Note that if U is dense open in X then U ∩Int(A)∩A is dense open in Int(A)∩A and U ∩Int(B)∩B is dense open in Int(B)∩B. We are now going to define a strategy σ for player two on A∪B. Suppose that in the first n innings, player two played the following sequence of dense open sets (U : i ≤ n). i If n = 2k for some k < ω, we let σ((U : i ≤ n)) = σ ((U ∩Int(A)∩A : i ≤ k)) i A 2i If n = 2k +1 for some k < ω, we let σ((U : i ≤ n)) = σ ((U ∩Int(B)∩B : i ≤ k)) i B 2i+1 Note now that, by the definition of σ and σ we have that: A B {σ((U : i ≤ n)) : n < ω} = i = {σ ((U : i ≤ k)) : k < ω}∪{σ ((U : i ≤ k)) : k < ω} = A 2i B 2i+1 = Int(A)∩A∪Int(B)∩B = X So σ is a winning strategy for player two in Gω(D ,D)) on X and we are done. (cid:3) 1 O In a similar way, one can prove the following propositions: SELECTIVE VERSIONS OF CHAIN CONDITION-TYPE PROPERTIES 9 Proposition 3.5. If player two has a winning strategy in Gω (D ,D) on A , for fin O i every i ≤ n then he also has a winning strategy in that game on A ∪A ∪···∪A . 1 2 n Proposition 3.6. The properties Sω(D ,D) and Sω (D ,D)) are preserved by finite 1 O fin O unions. The natural game version of the weak Rothberger property, introduced in the end of the previous section is the game Gω(O,O ). This is tightly connected to the dual 1 D version of the point-picking game studied by Berner and Juhász in [6]. Definition 3.7. Let Gp(κ) be the following game. At inning α < κ, player one picks o a point x ∈ X and player two picks an open set U such that x ∈ U . Player one α α α α wins if and only if U is dense in X. α<κ α We may call thisSthe open-picking game. We prove that the game Gκ(O,O ) and 1 D the game Gp(κ) are dual, in the following sense. o Theorem 3.8. (1) Player one has a winning strategy in Gp(κ) if and only if player two has a o winning strategy in Gκ(O,O ). 1 D (2) Player two has a winning strategy in Gp(κ) if and only if player one has a o winning strategy in Gκ(O,O ). 1 D Proof. The direct implication of (1) is easy to see. Indeed, let τ be a winning strategy for player one in Gp(κ). Given an open cover U let σ((U)) be any open set O ∈ U o such that τ(∅) ∈ O. Assuming we have defined σ for the first α many innings, and {O : β ≤ α} be a sequence of open covers, let σ((O : β ≤ α)) be any open set β β O ∈ O such that τ((σ(O : γ ≤ β) : β < α)) ∈ O. Then σ is a winning strategy α γ for player two in Gκ(O,O ). Indeed, let (O ,O ,O ,O ,...O ,O ,...) be a play, 1 D 0 0 1 1 α α where player two plays according to σ. Then τ((σ(O : γ ≤ β) : β < α)) ∈ O , and γ α hence {O : α < κ} is dense, as τ is a winning strategy for player one in Gp(κ). α o To prove the converse implication of (1), σ be a winning strategy for player two in S Gκ(O,O ) on some space X. 1 D Claim. Let (O : α < β) be a sequence of open covers. Then there is a point α x ∈ X such that, for every neighbourhood U of x there is an open cover U with U = σ((O : α < β)⌢(U)). α Proof of Claim. Recalling that O denotes the set of all open covers of X, let V = {V open: (∀U ∈ O)(V 6= σ((O : α < β)⌢(U))}. Its definition easily implies that V α cannot be an open cover, and hence there is a point x ∈ X \ V. By definition of V we must have that for every neighbourhood U of x there is an open cover U such that U = σ((O : α < β)⌢(U)) and hence we are done. S △ α 10 LEANDROF. AURICHI,SANTI SPADARO,ANDLYUBOMYRZDOMSKYY Use the Claim to choose a point x such that, for every neighbourhood U of x 0 0 there is an open cover U with σ(U) = U and let τ(∅) = x . 0 Suppose we have defined τ for the first α many innings. Let now {V : β ≤ α} β be a sequence of open sets and {O : β < α} be a sequence of open covers such β that V = σ((O : γ ≤ β)), for every β < α. Use the claim to choose a point β γ x such that, for every open neighbourhood U of x there is an open cover O with α α U = σ((O : β < α)⌢(O)) and let τ((V : β ≤ α)) = x . β β α We now claim that τ is a winning strategy for player one in Gp(κ). Indeed, let o (x ,V ,x ,V ,...x ,V ,...) be a play where player one uses strategy τ. Then there 0 0 1 1 α α must be a sequence of open covers {O : α < κ} such that V = σ((O : α < β)), for α β α every β < κ. Since σ is a winning strategy for two in Gκ(O,O ) then {V : α < κ} 1 D α is dense in X and this proves that τ is a winning strategy for player one in Gp(κ). o S To prove the direct implication of (2), let τ be a winning strategy for player two in Gp(κ). We define a winning strategy σ for player one in Gκ(O,O ) as follows: in his 0 1 D first move player one plays the open cover σ(∅) = {τ(x) : x ∈ X}. Assuming we have defined σ for every inning β, with β < α, let (U : β < α) be a sequence of open sets β such that there is a sequence {x : β < α} of points with U = τ(x : γ ≤ β). Then β β γ we simply define σ((U : β < α)) to be {τ((x : β < α)⌢(x)) : x ∈ X}. We claim β β that σ is a winning strategy for player one in Gκ(O,O ). 1 D Indeed, let (O ,U ,...O ,U ...) be a play of Gκ(O,O ) where player one plays 0 0 α α 1 D according to σ. Therefore, we can find a sequence {x : α < κ} of points, such that α U = τ((x : γ < α)), and hence U is not dense, since τ is a winning strategy α γ α<κ α for player two inGp(κ). So σ must be a winning strategy for player one inGκ(O,O ). o 1 D S To prove the converse implication of (2), let σ be a winning strategy for player one in Gκ(O,O ). We will use σ to define a winning strategy τ for player two in Gp(κ). 1 D o Given a point x ∈ X, let τ((x)) be any open set U ∈ σ(∅) such that x ∈ U. Now suppose τ has been defined for all sequences of points of ordinal length less than α. Given a sequence {x : β ≤ α} ⊂ X, let τ((x : β ≤ α)) be any open set β β U ∈ σ((τ((x : γ ≤ β) : β < α)) such that x ∈ U. We claim that τ thus defined is a γ α winning strategy for player two in Gp(κ). o Indeed, let x ,U ,x ,U ,...x ,U ... be a play of Gp(κ), where player two plays 0 0 1 1 α α o according to τ. Then U ∈ σ((τ((x : γ ≤ β) : β < α)), for every α < κ and hence, α γ since σ is a winning strategy for player I in Gκ(O,O ) we must have that U 1 D α<κ α is not dense, and we are done. S (cid:3) We are now going to exploit this result to give a short proof to a result of Angelo Bella from [7].

Selective versions of chain condition-type properties

Similar Selective versions of chain condition-type properties