κ-strong sequences and the existence of generalized independent families

Abstract In this paper we will show some relations between generalized versions of strong sequences introduced by Efimov in 1965 and independent families. We also show some inequalities between cardinal invariants associated with these both notions.


Introduction
The strong sequences method was introduced by Efimov in 1965 [1] as a useful tool for proving some famous theorems in dyadic spaces, (i.e. continuous images of Cantor cube). Among others Efimov proved that strong sequences do not exist in the subbase of Cantor cube. The problem of the existence of strong sequences in the other spaces was raised in 90's of the last century (see [2] or [3] for more historical details).
The problem of the existence of independent families was raised among others by Fichtenholz and Kantorovitch [4] and Hausdorff [5]. There was shown that for each cardinal Ä there exists an independent family of size 2 Ä . Engelking and Karłowicz introduced in [6] a generalization of the definition of this notion by considering partitions of a given set instead of a pair of sets. However, the name "generalized independent family" was used for the first time by Hu in [7]. In [8] Elser proved among others that under some assumptions there exists a generalized independent family (see Corollary 2.6.), using the result obtained by Hu (see Theorem 2.4 in [7]).
The aim of this paper is to show relations between generalized independent families and generalized strong sequences. There will be also shown relations between cardinal invariants concerning both mentioned notions.
The notation used in the paper is standard for the field and can be found in e. g. [9] and [10].

Generalized strong sequences
Let .X; r/ be a set with an arbitrary relation r and let Ä be a cardinal. By P .X / we denote the power set of X , i. e. the family of all subsets of X . We say that a set A X has a bound iff there exists b 2 X such that for all a 2 A we have .a; b/ 2 r. (If elements a; b 2 X have not a bound we say that they are incompatible). We say that a set A X is Ä directed iff each subset of A of cardinality less than Ä has a bound.
The following theorem is the generalization of Theorem 3 in [3]).
Theorem 2.2 (on Ä-strong sequences). Letˇ; Ä; ; be cardinals such that ! Äˇ , <ˇ, Ä < andˇ; be regular. Let X be a set of cardinality . If there exists a Ä-strong sequence fH˛ X W˛< g with jH˛j Ä 2 for all˛< , then there exists a Ä-strong sequence fT˛W˛<ˇg with jT˛j < Ä for all˛<ˇ.
Proof. Consider a Ä-strong sequence fH˛W˛< g as was done in the theorem. We define families fP˛ 2 P.X /W <ˇg with the following properties: (i) f˛ W <ˇg is an increasing subsequence of ; (ii) P˛ D fT g 2 OEH˛ <Ä W jf 1 ;h .T g /j D ; for some h 2 ı ; g 2 ; ı < g; (iii) f˛ ;h W A h ı n f˛ g ! OEH˛ <Ä is a function given by the formula f˛ ;h . / D T g ; h 2 ı ; g 2 ; ı < ; . We proceed by transfinite recursion. Take˛0 < and H˛0 . (Without the loss of generality we can assume that 0 D 0). By Definition 1 for each˛>˛0 there exists T 2 OEH˛0 <Ä such that T [ H˛is not Ä-directed. Consider a function f˛0 W n f˛0g ! OEH˛0 <Ä given by the formula f˛0 .
Then the function f˛0 determines a partition of n f˛0g into at most 2 < elements. Since is regular the family given by the formula the function f˛Á C1;g determines a partition of A g Á n f˛Á C 1g into at most 2 < elements. Since is regular the family is non-empty and fulfills (ii)-(iii). Now we assume that Á is limit and for all ı < Á the families P˛ı D fT g ı 2 OEH˛ı <Ä W jf 1 ı ;h .T g ı /j D ; for some h 2 ; g 2 ı ; < ıg fulfilling (ii)-(iii) have been defined. We set . Thus we have constructed at least one Ä-strong sequence of the form fT g 2 P˛ W <ˇ; g 2 g: Suppose now that at least one of defined above Ä-strong sequences has length >ˇ. Each set T g determines a set A g of cardinality , i. e. there is defined a function f˛ ;g as in (iii). Let D supfjP˛ jW < g. Then there would exist > pairwise disjoint sets A g of cardinality . A contradiction.
Corollary 2.3. Letˇ; Ä; be cardinals such that ! Äˇ , Ä < andˇ; be regular. Let X be a set of cardinality . Then either X contains a Ä-directed subset of cardinality or there exists a family of cardinalityˇconsisting of subsets of OEX <Ä with the property: for each A; Proof. Without the loss of generality we can assume that X Â . Suppose that each Ä-directed subset of X has cardinality less than . We will construct a Ä-strong sequence fH˛W˛< g via transfinite recursion.
Assume that for˛< the Ä-strong sequence fH Á X n S <Á H W Á <˛g has been defined. Since jH Á j < then j S Á<˛H Á j < and is regular we have jX n S Á<˛H Á j D . Now we will construct H˛. Let˛be successor. Let H˛ X n S Á<˛H Á be a maximal Ä directed set.
Let H˛be the next element of the Ä-strong sequence. By Theorem 2.2 there exists a Ä-strong sequence fT˛W˛< g such that jT˛j < Ä for all˛<ˇ. If T˛are not pairwise disjoint then we take the family consisting of sets T 0 such that T 0 0 D T 0 and T 0 D T˛n Sˇ<˛Tˇf or˛-successor and T 0 D sup. Sˇ<˛T 0 / for˛-limit. This completes the proof.
The next corollary follows immediately from Corollary 2.3.
Corollary 2.4. Letˇ; Ä; be cardinals such that ! Äˇ , Ä < andˇ; be regular. Let X be a set of cardinality . Then either X contains a Ä-directed subset of cardinality or there exists a subset of X of cardinalityˇconsisting of pairwise incompatible elements.
The next result in this paragraph will be the special case of previous ones for .P .X /; Â/.
A family A P .X / is closed under taking Ä -intersections i.e. for all A 0 A such that jA 0 j < Ä we have T A 0 2 A. Let Ä; be cardinals with Ä < . A family of sets A P .X /, with jAj , is called a Ä-vaulted family iff for each subfamily B A of cardinality less than Ä we have T B 6 D ;: Theorem 2.5. Letˇ; Ä; be cardinals such that ! Äˇ , Ä < andˇ; be regular. Let X be a topological space of cardinality . Let A P.X / be a family of sets with jAj D closed under taking Ä-intersections. Then A contains a Ä-vaulted family of cardinality or A contains a subfamily of cardinalityˇwhich consists of pairwise disjoint sets.
Proof. Let A D fA W < g be a family as is required in the theorem. Define a partial ordered set P D f < W A 2 Ag with the following relation.

Generalized independent families
In [7] the following definition was introduced Definition 3.1 ( [7]). Let I D ffIˇWˇ< ˛g W˛< g be a family of partitions of an infinite set S with each ˛ 2 and let Ä; ; Â be cardinals. If for any J 2 OE <Â and for any f 2 …˛2 J ˛t he intersection T fI f .˛/ W˛2 J g has cardinality at least Ä, then I is called a .Â; Ä/-generalized independent family on S. Moreover, if ˛D for all < , then I is called a .Â; Ä; /-generalized independent family on S .
Notice that an independent family considered in [6] is .!; 1; jSj/-independent family on a set S . Moreover in [8] and [7] there are shown some results concerning cardinality of generalized independent families. In Theorem 3.2 we show the relation between the existence of generalized independent families and Ä-strong sequences. Let A P .X /. Then c.A/ D supfjBjW B is a subfamily of pairwise disjoint sets of Ag: Theorem 3.2. Letˇ; Ä; be cardinals such that ! Äˇ , Ä < andˇ; be regular. Let X be a topological space of cardinality . Let A P.X / be a family of cardinality closed under taking Ä-intersections and such that c.A/ <ˇ. Then there exists a .Ä; 1/-generalized independent family of cardinality .
Proof. Consider a family P art D fP˛W jP˛j D ˛;˛< g of all partitions of . Since ˛Ä for all˛< , jP artj . By Theorem 2.5 there exists a Ä-vaulted family A P art of cardinality . We will construct a .Ä; 1/-generalized independent family via transfinite recursion.
Assume that for Á < the .Ä; 1/ generalized independent family I D fP˛2 AnfP W <˛gW˛< Ág has been defined. Clearly, I has the property that for any J 2 OEÁ <Ä and f 2 …˛2 J ˛t he intersection is a f .˛/-element of the partition P˛. Since Á < and jAj D , A n fP˛W˛< Ág 6 D ;. Hence we can continue our construction. Let I D fI W I D \ fI f .˛/ W˛2 J g for some J 2 OEÁ <Ä and some f 2 …˛2 J ˛g : Observe that S I D . If not, then there exists ı 2 n S I. It would mean that there is no J 2 OEÁ <Ä and f 2 …˛2 J ˛s uch that ı 2 T fI f .˛/ W˛2 J g: Then ı 6 2 P˛for some P˛2 I. A contradiction because P˛2 P art: Let Á be a successor. Let P Á 2 A n fP˛W˛< Ág be a partition with the property that for all I 2 I there exists I Á 2 P Á such that I I Á . For a set J 2 OE <Ä and f 2 …˛2 J ˛c hoose fI f .˛/ W˛2 J g. Since the family A is Ä-vaulted, T fI f .˛/ W˛2 J g 6 D ;: If Á is the limit, then we take P Á D T˛< Á P˛. Thus the .Ä; 1/ generalized independent family fP˛D fIˇWˇ< ˛g W˛< g has been defined.
In [6] it is proved the result related to the density of product of topological spaces (Theorem 8) using the theorem on the existence of independent families (Theorem 3). Following [8], we give the definition of Ä-box product.
Let Ä; be cardinals with @ 0 Ä Ä Ä and fX i g i 2 be a family of topological spaces. Then Ä i 2 X i denotes the Ä-box product which is induced on the full Cartesian product … i 2 X i by the canonical base where P <Ä . / WD fI W jI j < Äg: The next two corollaries follow from Theorem 2.5 and Corollary 2.3 and Theorem 4.3 in [7]. Corollary 3.3. Letˇ; Ä; be cardinals such that ! Äˇ , Ä < andˇ; be regular. Let X be a topological space of cardinality . Let A P.X / be a family of cardinality closed under taking Ä-intersections and such that c.A/ <ˇ. Let fX˛g˛< be a family of topological spaces such that d.X˛/ Ä ˛f or all˛< . Then d. ˛2Ä .X˛// Ä jS j.
A topological space X is irresolvable if X does not have disjoint dense subsets.
Let be a cardinal. An ideal I P .X / is called -complete if S˛< A˛2 I for˛< and A˛2 I. In [7], one can find the following lemma Lemma 3.4. Suppose .X; T / is an open-hereditarely irresolvable space and T is a P Â -topology for some regular cardinal Â . Let N denote the ideal of nowhere dense subsets, and let be the smallest cardinal such that N is not -complete. Then for any < C < and Á < Â , N is . Á / C -complete.
Corollary 3.5. Letˇ; Ä; Â; be cardinals such that ! Äˇ , Ä < and Â < withˇ; ; Â are regular. Let X be a topological space of cardinality . Let A P.X / be a family of cardinality closed under taking Ä-intersections and such that c.A/ <ˇ. Suppose that I is a family of partitions of a set S with each ˛< Â . Let N be the ideal of nowhere dense set of the simple topology induced by I and let be the smallest cardinal such that I is notcomplete. Then 1) there is a nonempty open set U of the simple topology such that U with the subspace topology satisfies all conditions in Lemma 3.4 and the ideal I U of nowhere dense set of U is -saturated and 2) 2 <Â D Â: We finish this paper by showing relations between cardinal invariants associated with both considered notions. Let Ä be a cardinal and let X be an infinite set of the cardinality Ä. Accept the following notations: O s Ä D supf˛W there exists a Ä-strong sequence in X of length˛g: i.Ä; 1/ D minf˛W there is no .Ä; 1/ generalized independent family on X of length˛g: Theorem 3.6. Let Ä; be cardinals such that Ä < and -regular and there exists a regular cardinalˇsuch that ! Äˇ . Let X be a topological space of cardinality . Then O s Ä Ä i.Ä; 1/: Proof. Without the loss of generality we can assume that X Â . Let I be a maximal .Ä; 1/-generalized independent family. If jIj D and I contains a Ä-vaulted family of cardinality then the theorem is complete. Suppose that each Ä-vaulted family has cardinality less than . (Then by Theorem 2.5 there are onlyˇpairwise disjoint sets in I for ! Äˇ ,ˇ; -regular). By transfinite recursion we will construct a Ä-strong sequence fH˛W˛< g. Assume that for <˛< the Ä-strong sequence fH W <˛g such that H X n S Á< H Á has been defined. Since jH j < , j S <˛H j < and is regular we have jX n S <˛H j D . Let˛be a sucessor. Hence there exists a maximal Ä-directed set H˛ X n S <˛H such that H˛[ H is not Ä-directed for all <˛.
If˛is limit, then H˛D S <˛H [ fxg, where x D min.X n S <˛H /. Obviously, H˛[ H is not Ä-directed for all <˛. Let H˛be the next element of the Ä-strong sequence. The proof is complete.
The easy consequence of Theorem 3.6 and Theorem 3.2. in [7] is Corollary 3.7. Let Ä; be cardinals such that Ä < and -regular and there exists a regular cardinalˇsuch that ! Äˇ . Let X be a topological space of cardinality . Then the following statements are equivalent: