HuAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 7, No. 2, 2006 pp. 203-209 Generalized independent families and dense sets of Box-Product spaces Wanjun Hu Abstract. A generalization of independent families on a set S is introduced, based on which various topologies on S can be defined. In fact, the set S with any such topology is homeomorphic to a dense sub- set of the corresponding box product space (Theorem 2.2). From these results, a general version of the Hewitt-Marczewski-Pondiczery theorem for box product spaces can be established. For any uncountable regular cardinal θ, the existence of maximal generalized independent families with some simple conditions, and hence the existence of irresolvable dense subsets of θ-box product spaces of discrete spaces of small sizes, implies the consistency of the existence of measurable cardinal (Theo- rem 4.5). 2000 AMS Classification: 03E05, 05D05; Secondary: 54A25. Keywords: Generalized independent family, box product. 1. Introduction Following notation in [5], a (θ, κ)-independent f amily on S is a subfamily I⊆ P(S) such that for any two disjoint subfamilies I0, I1 ⊆ I with |I0∪I1| < θ, the set ⋂ {A : A ∈ I0} ∩ ⋂ {S \ A : A ∈ I1} has cardinality κ. Given a space 〈X, T 〉, it is irresolvable ([9]) if X does not have two disjoint dense subsets. Following [3], let S(〈X, T 〉) be the smallest cardinal κ such that every family of pairwise disjoint nonempty open sets has size strictly less than κ. Please refer to [10] about cardinals and ideals, and [6] and [3] for topological terminologies. The Hewitt-Marczewski-Pondiczery theorem and Hausdorff’s theorem (i.e., there are uniformly independent families of size 2κ on any set S of size κ. See [8], [6] for more details) are equivalent, since each separated (θ, κ)-independent family of size 2|S| on a set S induces a Tychonoff topology on S which is homeomorphic to a dense subset of {0, 1}2 |S| . Such kind of topologies induced 204 W. Hu by independent families appeared in a different form in van Douwen’s paper [4] and then the paper [5] by F.W. Eckertson. On the other hand, Kunen [11] established the equiconsistency between the existence of maximal σ-independent family and the existence of measurable cardinals. Later Kunen, Szymanski and Tall in [12] (see also [14]) studied the properties of the ideal of nowhere dense subsets of a λ-Baire irresolvable space, and also gave a method to construct a λ-Baire open-hereditarily irresolvable (the term ”strongly irresolvable” was used. We follow the notation in [4]) topology from a λ-complete ideal with a lifting. In [14], it was shown that a λ-complete ideal on λ with certain conditions has a lifting. In this paper, we study a generalization of independent families and its relation to box product spaces. We provide a generalization of the equivalence between Hausdorff’s theorem and the Hewitt-Marczewski-Pondiczery theorem to generalized independent families and dense subsets of box product spaces (Theorem 3.2) (See also [7]). We show, in Section 2, that various topologies can be defined on a set S by any generalized independent family on S, and any such topology is homeomorphic to a dense subset of the corresponding θ-box product spaces. This general equivalence enables us to obtain similar work (Section 4) like that in [11] and [12] by substituting Baire irresolvable dense subsets with irresolvable dense subsets of box product spaces. 2. Generalized independent families and induced topologies An independent family can be viewed as a family of partitions on some set S, in which each partition consists of two subsets. In general, we can consider the following generalized version. Definition 2.1. Let I= {{Iβα : β < λα} : α < τ} be a family of partitions on an infinite set S with each λα ≥ 2, and let κ, λ, θ ≥ ω be three cardinals. • If for any J ∈ [τ ]<θ and any f ∈ Πα∈J λα the intersection ∩{I f (α) α : α ∈ J} has size at least κ, then I is called a “(θ, κ)- generalized independent family” on S, and a “(θ, κ, λ)-generalized in- dependent family” when λα = λ for all α < τ . • I is called “separated” if for any {x, y} ∈ [S]2, there exists an α < τ and β < λα such that x ∈ I β α and y /∈ I β α . A (θ, κ, 2)-generalized independent family is a (θ, κ)-independent family, and a σ-independent family defined in [11] is an (ω1, 1)-independent family. Let I= {{Iβα : β < λα} : α < τ} be a (θ, κ)-generalized independent family on some infinite set S, and let {〈Xα, Tα〉 : α < τ} be a family of topological spaces such that |Xα| = λα for each α < τ . For each α < τ , index the α-th partition of I by {Ixα : x ∈ Xα}, and for each nonempty open subset U ∈ T α, define BUα = ⋃ {Ixα : x ∈ U}. Set Bα:= {B U α : ∅ 6= U ∈ Tα}. The family Bα is a sub-base for a topology on S. We denote it by SXα , and we use I{Xα} to denote the topology generated by {SXα : α < τ}. When each 〈Xα, Tα〉 is discrete, the topology I{Xα} is called “the simple topology” induced by I. Generalized independent families and dense sets of Box-Product spaces 205 It is clear that 〈S, I{Xα}〉 is a Pθ-space whenever θ is regular. The space is Hausdorff if I is separated and each 〈Xα, Tα〉 is Hausdorff, and zero-dimensional if in addition each 〈Xα, Tα〉 is zero-dimensional. In the rest of this section, we only consider Hausdorff spaces and separated families. Theorem 2.2. Let I and {〈Xα, Tα〉 : α < τ} be as above. Any space 〈S, I{Xα}〉 is homeomorphic to a dense subset of 2τθ 〈Xα, T α〉 Proof. For each α < τ , define fα : 〈S, I{Xα}〉 → 〈Xα, T α〉 such that fα(I x α) = x. By our definition of I{Xα}, we know that fα is a continuous map. Since I is separated, the family {fα : α < τ} separates points in 〈S, I{Xα}〉. Consider the map f = ∆α<τ fα : 〈S, I{Xα}〉 → 2 τ θ 〈Xα, T α〉 such that f (s) = {fα(s)}α<τ for all s ∈ S. Certainly f is a one-one map, and f separates points. We need to show the following: (1) f is continuous; (2) the range of f is dense in 2τθ 〈Xα, T α〉; (3) f separates points and closed sets. To see that f is continuous, it is enough to show that for any set A ∈ [τ ]<θ and any family {∅ 6= Uα ∈ T α : α ∈ A} of nonempty open sets, the pre-image of the corresponding open set of 2α∈AUα is open. By the definition of f , a point s ∈ S is in the pre-image of that open set if and only if fα(s) ∈ Uα for all α ∈ A, and fα(s) ∈ Uα if and only if there exists some x ∈ Uα such that s ∈ Ixα ⊆ B Uα α . Hence s ∈ ⋂ {BUαα : α ∈ A} ∈ I {Xα}. Therefore the pre-image of 2α∈AUα is ⋂ {BUαα : α ∈ A}, which is open in I{Xα}. For (2), we need to show that there exists a point s ∈ S such that f (s) is in the corresponding open set of 2α∈AUα. Using the same argument as above, it is enough to show that ⋂ {BUαα : α ∈ A} is nonempty. Since I is a θ-generalized independent family, it is clear that ⋂ {BUαα : α ∈ A} 6= ∅. Hence the range of f is dense in 2τθ 〈Xα, T α〉. It remains to show that f separates points and closed sets. Let s be a point and let F be a closed subset in 〈S, I{Xα}〉 such that s /∈ F . Since B is a base for I{Xα}, for some set A ∈ [τ ] <θ and some family {∅ 6= Uα ∈ T α : α ∈ A}, we have s ∈ ⋂ {BUαα : α ∈ A} ⊆ F c. Obviously f (s) is in the corresponding open set of 2α∈AUα. We show that the corresponding open set of 2α∈AUα is disjoint from f (F ). Since the projection into any |A| < θ many products is open and continuous, it suffices to prove that 2α∈AUα∩ ∆α∈Afα(F ) = ∅. But this can be proved by the same argument used before: if fα(t) ∈ Uα for some t ∈ F , then t ∈ Ixα for some x ∈ Uα and hence t ∈ B Uα α , which implies that t ∈ ⋂ {BUαα : α ∈ A} ⊆ F c, contradicting our early assumption. Since {f} is continuous, separates points, and separates points and closed sets, it is a homeomorphism onto its range. It maps 〈S, I{Xα}〉 onto a dense subset of 2τθ 〈Xα, T α〉. � The following corollary is clear. Corollary 2.3. Let I= {{Iβα : β < λα} : α < τ} be a (θ, κ)-generalized independent family on S, and let {〈Xα, T α〉 : α < τ} be a family of topological spaces such that d(〈Xα, T α〉) ≤ λα for all α < τ . Then d(2 τ θ 〈Xα, T α〉) ≤ |S|. 206 W. Hu The converse of Theorem 2.2 can be established for box product spaces of discrete spaces. Theorem 2.4. For any dense subset D in 2τθ D(λα), there exists a (θ, 1)- generalized independent family I on D. The set D is irresolvable if and only if I is a maximal (θ, 1)-generalized independent family. 3. The Hewitt-Marczewski-Pondiczery theorem for box product spaces Definition 3.1. Let κ, θ, λ be two cardinals with κ, θ infinite. Let S be an infinite set of size κ. The cardinal i(κ, θ, λ) is the smallest cardinal τ such that there are no (θ, 1, λ)-generalized independent families on S of size τ . The following generalizes the Hewitt-Marczewski- Pondiczery theorem. Theorem 3.2. Let S be a set and let θ, τ, λ be three cardinals with θ infinite. Then the following are equivalent. • τ < i(|S|, θ, λ). • d(2τθ 〈Xα, Tα〉) ≤ |S| holds for any family of topological spaces {〈Xα, Tα〉 : α < τ} with each d(Xα) ≤ λ. Proof. (1)→(2). By Corollary 2.3. (2)→ (1). Let D be a dense subset of 2 τ θ D(λ) such that |D| = |S|. For each α < τ and β < λ, let I β α = D ∩{{xζ}ζ<τ ∈ 2τθ D(λ) : xα = β}. Then the family I= {{I β α : β < λ} : α < τ} is a (θ, 1, λ)- independent family on D. Hence there is a (θ, 1, λ)-independent family of size τ on S, which implies τ < i(|S|, θ, λ). � Comfort and Negrepontis in [2] showed that |S|<θ = |S| is equivalent to the statement that there exists a subfamily of SS of size 2|S| that is of θ-large oscillation, which implies the existence of a (θ, 1, |S|)-independent family of size 2|S| on S. On the other hand, assuming there exists a (θ, 1, |S|)-independent family I of size 2|S| on S, for each β < 2|S| let fβ : S → S be such that fβ(I s β ) = s for each s ∈ S. Then the family {fβ : β < 2 |S|} is a family of θ-large oscillation. Hence, we have the following theorem. Theorem 3.3. i(|S|, θ, |S|) = (2|S|)+ if and only if |S|<θ = |S|. We show in the following theorem that, in general, the cardinal i(|S|, θ, |S|) is regular. Theorem 3.4. Let θ, λ be two infinite cardinals such that θ ≤ λ. Then i(λ, θ, λ) is regular. Proof. Let τ < i(λ, θ, λ) and let {τα : α < τ} be cardinals such that τα < i(λ, θ, λ). Let also µ = sup{τα : α < τ}. By Theorem 3.2, for each α < τ , the box product 2τα θ λ has density λ. By Theorem 3.2 again, the space 2 µ θ λ = 2 τ θ (2 τα θ λ) has density λ. Hence µ < i(λ, θ, λ) according to Theorem 3.2. � It is clear that for any infinite set S and two cardinals λ1 ≤ λ2, we have (2|S|)+ ≥ i(|S|, θ, λ1) ≥ i(|S|, θ, λ2). When |S <θ| = |S|, we have i(|S|, θ, λ) = i(|S|, θ, |S|) = (2|S|)+ for any λ < |S|. Generalized independent families and dense sets of Box-Product spaces 207 4. Maximal generalized independent families The simple topology induced by a maximal (θ, 1)-generalized independent family is irresolvable. Similarly, the simple topology induced by a maximal (θ, 1, λ)-independent family is λ-irresolvable. In this section, we show that for any uncountable regular cardinal θ, the existence of maximal (θ, 1)-generalized independent families with some simple conditions (equivalently, the existence of irresolvable dense subsets of θ-box product spaces with some simple conditions) implies the consistency of the existence of measurable cardinals. Lemma 4.1. Suppose 〈X, T 〉 is an open-hereditarily irresolvable space and T is a Pθ-topology for some regular cardinal θ. Let N denote the ideal of nowhere dense subsets, and let λ be the smallest cardinal such that N is not λ-complete. Then for any γ < γ+ < λ and β < θ, N is (γβ )+-complete. Proof. Since the topology is open hereditarily irresolvable, N = {A ⊆ S : Ao = ∅}. For a contradiction, let us assume that there ex- ists Yf ∈ N for each f ∈ γ β such that the Yf are disjoint and ⋃ f Yf ⊇ U for some nonempty open set U . We claim that there exists some member Yg /∈ N . Inductively define g : β → γ and a decreasing chain of non-empty basic open sets {U ζ : ζ < β} so that • U 0 = U , • U ζ = ⋂ {U η : η < ζ}, • U ζ+1 ⊆ U ζ and U ζ+1 ⊆ ⋃ {Yf : f (ζ) = g(ζ)}. When ζ < θ is a limit, the set ⋂ {U η : η < ζ} defined in (2) is a non- empty open set, since T is a Pθ-topology. For (3), we have γ-many disjoint sets {Nα = ⋃ {Yf : f (ζ) = α} : α ∈ γ}. The union of these sets contains U and hence U ζ . Since the topology is open hereditarily irresolvable and N is γ+-complete, one of these sets {Nα ∩ U ζ : α < γ}, say Nα ∩ U ζ , has non-empty interior U ζ+1. Set g(ζ + 1) = α. We have ⋂ {U ζ : ζ < β} ⊆ ⋂ ζ<β ⋃ {Yf : f (ζ) = g(ζ)} = Yg contradicting Yg ∈ N . � In [2], Comfort and Negrepontis introduced the notion of strongly θ-inaccessible: a cardinal λ is called strongly θ-inaccessible if βγ < λ whenever β < λ and γ < θ. Given a cardinal θ, denote by θin the smallest cardinal λ such that λ is strongly θ-inaccessible. Lemma 4.2. Let everything be as in Lemma 4.1. Then • λ is regular; • if λ is a successor cardinal, say λ = λ′+, then λ′ is strongly θ-inaccessible. • if λ is a limit cardinal, then λ is strongly θ-inaccessible Proof. (1) is trivial. If λ = λ′+, then for any γ < λ′ and β < θ, (γβ)+ ≤ λ′ (Lemma 4.1), and hence (γβ ) < λ′. This gives (2). (3) is trivial. � Let everything be as in Lemma 4.1. Let us assume further that S(〈X, T 〉) ≤ λ with λ defined in Lemma 4.1. Then it is easy to see that the ideal N is 208 W. Hu λ-saturated. Under these assumption, there exists a λ-saturated (hence λ+- saturated) λ-complete ideal over λ (using the proof of Lemma 27.1 in [10]). Lemma 35.10 and Theorem 86 in [10] show that λ is a measurable cardinal in some model of ZFC. In the following we show that for any uncountable regular cardinal θ, if there exists a maximal (θ, 1)-generalized independent family with some conditions, then the induced simple topology satisfies above conditions. Theorem 4.3. Let θ be a regular cardinal. Suppose there exists a maximal (θ, 1)-generalized independent family I= {{Iβα : β < λα} : α < τ} on a set S with each λα < θin. Let N be the ideal of nowhere dense set of the simple topology induced by I and let λ be the smallest cardinal such that I is not λ-complete. Then • there is a nonempty open set U of the simple topology such that U with the subspace topology satisfies all conditions in Lemma 4.1 and the ideal IU of nowhere dense set of U is λ-saturated; and • 2<θ = θ Proof. (i) Let 〈S, T 〉 be the simple topology induced by I. Since I is a maxi- mal (θ, 1)-generalized independent family, it is irresolvable. Using a standard argument ([9]), there is a nonempty basic open set U the subspace topology on which is hereditarily irresolvable. Let NU be the set of all nowhere dense subsets in 〈U, T 〉, i.e, the set U with the subspace topology inherited from T . By Lemma 4.2, if λ is a limit cardinal, then λ is strongly θ-inaccessible. If λ is a successor cardinal, say λ = λ′+, then λ′ is strongly θ-inaccessible. Using Theorem 2.3 in [2], we have that S(〈S, T 〉), and hence S(〈U, T 〉), is ≤ λ if λ is a limit cardinal, and < λ otherwise. Hence NU is λ-saturated. (ii) The proof here uses a similar argument as that of Lemma 1.4 in [11]. For each θ′ < θ we produce a map from θ onto 2θ ′ . Index θ as A × B with A = {aη : η < θ} and B = {bζ : ζ < θ ′}. Consider the family {I0α : α < θ} = {I 0 aη bζ : η < θ, ζ < θ′}. For each x ∈ X, define φx : θ → 2 θ ′ so that φx(η)(ζ) = 1 if and only if x ∈ I 0 aη bζ . For each f ∈ 2θ ′ , let Rf be {x ∈ X : f /∈ range(φx)} = {x ∈ X : f 6= φx(η) for all η < θ}. We show that ⋂ f∈2θ ′ (X \ Rf ) 6= ∅ by proving Rf ∈ N and applying Lemma 4.1, which shows that for some x, φx is onto. Suppose that Rf contains U = ⋂ {U σ(α) α : α ∈ A}, a non-empty basic open set, for some set A ∈ [τ ]<θ and some σ ∈ Πα∈Aλα. Then there is an η < θ such that A ∩ ∪{(aη, bζ ) : ζ < θ ′} = ∅. Now consider the open set U ′ = U ∩ {I0aη bζ : ζ < θ ′ and f (ζ) = 1} ∩{S \ I0aη bζ : ζ < θ ′ and f (ζ) = 0}. It is clear that U ′ 6= ∅ and U ′ ⊆ {s ∈ S : φs(η) = f} ∩ U ⊆ (S \ Rf ) ∩ U , a contradiction. � Generalized independent families and dense sets of Box-Product spaces 209 The following theorem is a direct corollary of Theorem 4.3. Corollary 4.4. For any uncountable regular cardinal θ, the existence of a maximal (θ, 1)-generalized independent family I= {{Iβα : β < λα} : α < τ} on a set S with each λα < θin implies the consistency of the existence of measurable cardinals, and 2<θ = θ. The corresponding conclusion is about the existence of irresolvable dense subsets in a θ-box product space. Theorem 4.5. Let θ be an uncountable regular cardinal, and let {λα ≥ 2 : α < τ} be a family of cardinals with each λα < θin. If there exists an irresolvable dense subset S of the θ-box product space 2 τ θ D(λα), then • It is consistent that there exists a measurable cardinal; and • 2<θ = θ. 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Math. 117 (1983), 75–80. [12] K. Kunen, A. Szymanski and F. Tall, Baire resolvable spaces and ideal theory, Prace Nauk., Ann. Math. Sil. 2(14) (1986), 98–107. [13] K. Kunen and F. Tall, On the consistency of the non-existence of Baire irresolvable spaces, http://at.yorku.ca/v/a/a/a/27.htm, 1998. [14] S. Shelah, Baire irresolvable spaces and lifting for a layered ideal, Topology Appl. 33 (1989), 217–231. Received May 2005 Accepted July 2006 Wanjun Hu (Wanjun.Hu@asurams.edu) Department of Mathematics and Computer Science, Albany State University, Albany GA 31705.