Disjoint Families Research Articles

Constructions of Some Non-σ-porous Sets on the Real Line

(i) if a Borel set A does not belong to A, then A+A contains an interval, and (ii) each disjoint family of Borel sets not belonging to A is countable, it is natural to ask if these properties also hold with A replaced by the class of σ-porous sets. These problems were posed by P. D. Humke [3] and W. Wilczynski at the symposium “Real Analysis” held in August 1982 in Esztergom, Hungary. J. Foran and P. D. Humke [2] showed some “enveloping” properties of σ-porous sets and posed a problem whether there exists a porous set contained in no σ-porous Gδ set. Here we give positive answer to the last question and prove that the class of σ-porous sets has neither of the properties (i) or (ii) even for perfect sets. To construct the corresponding examples we give a general method of the construction of perfect non-σ-porous sets, a special case of which has been used by L. Zajicek [4] in his construction of perfect non-σ-porous set of measure zero. For a subset S of the real line we define the set

Cutting rectangles in the plane

Let F be a planar family of pairwise disjoint translates of a parallelogram. It is shown that if every four members of F are intersected by some straight line, then there is a straight line which intersects all but at most two members of F . If it is instead assumed that every two members of F are intersected by some line that is parallel to one of the edges of the parallelogram, then the same conclusion holds.

Point-finite Borel-additive families are of bounded class

We prove the following theorem, which answers a question originally raised by J. Kaniewski and R. Pol: THEOREM. If 9 is a point-finite family of subsets of a metric space X such that the union of every subfamily is a Borel set of X, then there exists a fixed countable ordinal a such that each member of 9 is a Borel set of class a in X. The proof is given in the general setting of abstract spaces. An application is made to the study of selectors for compact-valued mappings and to the Borel measurability of graphs. In their joint paper [5], J. Kaniewski and R. Pol raised the question of whether a point-finite Borel-additive family in a metric space has the property that, for some fixed a < xl, all members of the family are of class a (cf. [5, Question 3, p. 1049], and also [1, Question 5, p. 315] and [4, Problem 10, p. 482]). For disjoint families, D. Preiss [7] has shown that such a bound exists, even in the more general setting of abstract measurable spaces (see the paper by W. Fleissner [1] for a short and elegant proof of Preiss's theorem). The purpose of this note is to prove that Preiss's result also holds in the point-finite case, and that this cannot be extended to the case of point-countable families.' Let X be any set, and let e be a family of subsets of X. We define inductively the family Ia Secondary 54C60, 54C65.

Mad Families and Ultrafilters

For each almost disjoint family X let $F(X) = \{ a \subseteq \omega :{\text {card}}\{ s \in X:s\backslash a\;{\text {is}}\;{\text {finite}}\} = {2^\omega }\} ,I(X) = \{ a \subseteq \omega :{\text {card}}\;\{ s \in X:{\text {card}}\;(s \cap a) = \omega \} = {2^\omega }\}$ . Assuming $P({2^\omega })$ we show that for each nonprincipal ultrafilter p there exist a maximal almost disjoint family X and an almost disjoint family Y with $F(X) = I(Y) = p$.

Mad families and ultrafilters

For each almost disjoint family X let F ( X ) = { a ⊆ ω : card { s ∈ X : s ∖ a is finite } = 2 ω } , I ( X ) = { a ⊆ ω : card { s ∈ X : card ( s ∩ a ) = ω } = 2 ω } F(X) = \{ a \subseteq \omega :{\text {card}}\{ s \in X:s\backslash a\;{\text {is}}\;{\text {finite}}\} = {2^\omega }\} ,I(X) = \{ a \subseteq \omega :{\text {card}}\;\{ s \in X:{\text {card}}\;(s \cap a) = \omega \} = {2^\omega }\} . Assuming P ( 2 ω ) P({2^\omega }) we show that for each nonprincipal ultrafilter p there exist a maximal almost disjoint family X and an almost disjoint family Y with F ( X ) = I ( Y ) = p F(X) = I(Y) = p .

Generalizations of almost disjointness, c-sets, and the baire number of βN-N

We consider certain combinatorial problems and their consequences in βN-N. Let F be any family of infinite subsets of N. Then we are interested in conditions under which it is possible to find an almost disjoint family A = { A F: Fϵ F } such that A ϵ⊆ F. If such a family A exists, we call F separable. We note that any family of cardinality less than c is separable, but that it is independent of the negation of the Continuum Hypothesis as to whether or not a union of ℵ 1 separable families is separable. We consider the separability of unions of ultrafilters and use our results to show that it is consistent that if S is any nowhere dense subset of βN-N, then there exists a family of c pairwise disjoint open sets each of which is disjoint from S but contains S in its closure. We also consider rectangular arrays of subsets of N subject to the condition that each row and each column is an almost disjoint family, and we consider various maximality questions concerning these. We then note that if there exists such an array with K rows and K + columns and the columns are all maximal, then βN-N may be covered by K + nowhere dense sets.

Static spherical configurations of cold matter in the Einstein-Cartan theory of gravitation

We investigate static, spherical configurations of cold catalized matter in the Einstein-Cartan theory of gravitation. Assuming that density of spin is proportional to the number density of baryonsn and using an equation of state of a degenerate, relativistic Fermi gas, we numerically integrated the relativistic equation of equilibrium. We have also studied the stability of those configurations. Configurations with central number densitync such that\(d\tilde p|dn|_{n = n_c } > 0\) where\(\tilde p\) is the effective pressure, are very similar to general relativistic configurations with the same central density. In the Einstein-Cartan theory there exists another disjoint family of equilibrium configurations for which\(d\tilde p|dn|_{n = n_c } 0\). Those configurations have very small masses ∼10−6 g and raddi ∼10−34 cm and are unstable.

Families of sets whose pairwise intersections have prescribed cardinals or order types Corrigenda

(i) J. Baumgartner has kindly drawn our attention to the fact that Theorem 2 as stated in (1) is false. A counter example is the case in which m = ℵ2; n = ℵ1; p = ℵ0. For by reference (3) of the paper (1) there is an almost disjoint family (Aγ: γ &lt; ω1) of infinite subsets of ω̲ Put Aν = ω̲ for ω1 ≤ ν &lt; ω2. Then, contrary to the assertion of that theorem, all conditions of Theorem 2 are satisfied. However, Theorem 2 becomes correct if the hypothesisis strengthened toIn fact, Baumgartner has proved the desired conclusion under the weaker hypothesis

Right zero unions of N-semigroups

=1 commutative semigroup is called archimedean if every element divides a positive integer power of each other element. An N-semigroup is one of four possible types of archimedean semigroups. Chrislock [l] extends the notion of archjmedeanness to noncommutative semigroups. An N*-semigroup is a noncommutative archimedean semigroup that can be thought of as a generalization of an N-semigroup. A basic reference on N-semigroups is [5]. Let F = {S a : u E R} be a disjoint family of semigroups. F has a right zero union, if there exists a semigroup T that is a disjoint union of the S, , where each S, is a left ideal of T. Thus, T is homomorphic onto the right zero semigroup R by mapping each S, to 01. It will be shown that an N*-semigroup is a right zero union of N-semigroups and conversely. Section 2 can be thought of as preliminary material. Section 3 examines right zero unions of N-semigroups. It will be shown in that section that a right zero union of N-semigroups is isomorphic to a subdirect product of an ,V-semigroup and a right zero semigroup. Conversely, any such subdirect product is a right zero union of N-semigroups. Section 4 proves, biter alia, the equivalence of the two concepts N*-semigroups and right zero unions of N-semigroups. An N-semigroup has a representation involving an abelian group G and a function I defined on G x G. Similarly, an N*-semigroup has a representation with an abelian right group H (if H = G x R then G is abelian) and an I* function defined on H x H. Section 4 will examine I* functions, and will conclude with a method for constructing them. Omitted proofs can be found in [4]. A g eneral reference to semigroup theory is [3]. The study of right zero unions is a special case of the study of “bands of semigroups.”

Mutually disjoint families of 0–1 sequences

Obtain a family of infinite binary sequences by fixing k positions each with 0 or 1 and allowing the nonfixed positions to vary over {0,1}. The object of this note is to find the maximum number of pairwise disjoint families and to determine the maximum possible number of distinct fixed positions. Extensions to n-ary infinite sequences will also be discussed.

R. M. Solovay and S. Tennenbaum. Iterated Cohen extensions and Souslin's problem. Annals of mathematics, ser. 2 vol. 94 (1971), pp. 201–245.

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a dense subset. (One shows first that the dense is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the countable dense subset condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2

Iterated Cohen Extensions and Souslin's Problem

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a dense subset. (One shows first that the dense is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the countable dense subset condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2

Some Sufficient Conditions for Maximal-Resolvability(1)

A topological space X is called maximally resolvable if it admits a largest possible family of pairwise disjoint, “maximally dense” subsets. More precisely, if Δ(X) denotes the least among the cardinal numbers of the nonvoid open subsets of X, then X is maximally resolvable if it has isolated points or there exists a family {Rα}α &lt; Δ(X) of subsets of X, called a maximal resolution for X, such that ∪{Rα | α &lt; Δ(X)} = X, Rγ ∩ Rδ==ϕ if γ ≠ δ, and, for each a and each nonvoid open subset V of X, the cardinality of Rα ∩ V is not less than Δ(X).

Correction to the paper "On relatively disjoint families of measures, with some applications to Banach space theory" this volume pp. 13-36

Correction to the paper "On relatively disjoint families of measures, with some applications to Banach space theory" this volume pp. 13-36

On relatively disjoint families of measures, with some applications to Banach space theory

On relatively disjoint families of measures, with some applications to Banach space theory

Intersections by hyperplanes

LetA be a finite nonempty family of nonempty disjoint closed and bounded sets in a Banach spaceE which is either separable and the conjugate of some Banach spaceX (i.e.E=X*) or, reflexive and locally uniformly convex. IfC denotes the weak*-closed convex hull of ∪ {A:A ∈A} then the set of points inE ∼C through which there is no hyperplane intersecting exactly one member ofA is discrete (or empty).

Further results on order types and decompositions of sets

In this paper the study of problems P and Q [3; 4] is continued. The reader is referred to the aforementioned two references for all unfamiliar terms and symbols. The first three sections deal with the decomposition of a linear set into a family of pairwise disjoint sets, the order types of the sets being required to satisfy some specified conditions. ?1 is concerned with the decomposition of an arbitrary linear set, of power 2H0, into a family of pairwise disjoint sets, the order types of the sets being pairwise incomparable. ?2 is concerned with the decomposition of certain linear sets into families of pairwise disjoint, similar sets. ?3 is concerned with the decomposition of an arbitrary linear set into families, of power No and 2V0, of pairwise disjoint sets, each set having property A. Let {I }, t <0, be a sequence of order types, of power 2 o each, such that at <X for each t. In ?4 it is shown that (1) problem P, as applied to ot and , =X, admits of a solution rt such that the rt are pairwise incomparable order types (Theorem 4.1); and (2) problem P, as applied to each o= 0 and /j = at, admits of a solution rt such that the rt are pairwise incomparable order types (Theorem 4.3). 1. Decompositions into incomparable order types. In this section the decomposition of a linear set into a finite number and into a denumerably infinite number of pairwise disjoint sets Ai, where the order types of the Ai are pairwise incomparable, is studied. The decomposition of a linear set into 2Ho pairwise disjoint sets Ai, where the order types of the Ai are pairwise incomparable, is treated in ?3 (Theorem 3.4). DEFINITION. A linear set E, of power 2H0, will be said to have property C if each element of E is a c-condensation point of E. For any linear set E, by K(E) is meant the set of similarity transformations of E into R. By K*(E) is meant the set K(E) {I}, where I is the identity transformation of E. For any similarity transformation f of A into B, by f* is meant the inverse of f.