Meager Sets Research Articles

On a rectangle inclusion problem

We show that if every set of reals of size 2 ℵ 0 {2^{{\aleph _0}}} contains a meager-to-one continuous image of a set that cannot be covered by less than 2 ℵ 0 {2^{{\aleph _0}}} meager sets, then there exists a null (Lebesgue measure zero) subset of the plane R × R \mathbb {R} \times \mathbb {R} that meets every nonnull rectangle X × Y X \times Y . The antecedent is satisfied, e.g., if ω 2 {\omega _2} Cohen reals are added to a model of the continuum hypothesis.

On the covering and the additivity number of the real line

We show that the real line R cannot be covered by k many nowhere dense sets iff whenever D = { D i : i ∈ k } D = \{ {D_i}:i \in k\} is a family of dense open sets of R there exists a countable dense set G of R such that | G ∖ D i | > ω |G\backslash {D_i}| > \omega for all i ∈ k i \in k . We also show that the union of k meagre sets of the real line is a meagre set iff for every family D = { D i : i ∈ k } D = \{ {D_i}:i \in k\} of dense open sets of R and for every countable dense set G of R there exists a dense set Q ⊆ G Q \subseteq G such that | Q ∖ D i | > ω |Q\backslash {D_{i}}| > \omega for all i ∈ k i \in k .

Baire Paradoxical Decompositions Need at Least Six Pieces

We show that in certain cases paradoxical decompositions of compact metric spaces using sets (or even [0, 1 ]-valued functions) with the property of Baire modulo meager sets need more pieces than paradoxical decompositions with unrestricted pieces. In particular, any Baire paradoxical decomposition of the sphere ${S^2}$ using isometries needs at least six pieces.

Baire paradoxical decompositions need at least six pieces

We show that in certain cases paradoxical decompositions of compact metric spaces using sets (or even [0, 1 ]-valued functions) with the property of Baire modulo meager sets need more pieces than paradoxical decompositions with unrestricted pieces. In particular, any Baire paradoxical decomposition of the sphere S 2 {S^2} using isometries needs at least six pieces.

A very discontinuous Borel function

AbstractIt is shown to be consistent that the reals are covered by ℵ1, meagre sets yet there is a Baire class 1 function which cannot be covered by fewer than ℵ2, continuous functions. A new cardinal invariant is introduced which corresponds to the least number of continuous functions required to cover a given function. This is characterized combinatorially. A forcing notion similar to, but not equivalent to, superperfect forcing is introduced.

Lower Density Topologiesa

AbstractBy considering lower density operators and their induced topologies in a general setting, some results of S. Scheinberg and E. Ľazarow et al. are unified and generalized. It is also shown that every σ‐finite complete measure space (X, M, m) has a lower density operator and that every such operator induces a topology making X a category measure space in the sense of J. C. Oxtoby, except that the measure need not be finite. One consequence is that category σ‐finite measure spaces must have the countable chain condition. Also, for every topological space (X, |Gt), there is a lower density operator on the σ‐field of sets having the property of Baire (relative to the σ‐ideal of meager sets). Further, in both the “measure” and “category” contexts, all induced lower density topologies have simple form. Finally, it is shown that the deep. J‐density operator on the σ‐field of subsets of the real line having the property of Baire is not a lower density operator.

Baire property and axiom of choice

We show that without using inaccessible cardinals it is possible to get models of “ZF+all sets of reals have the Baire property +DC(ω1)” and “ZFC+all projective sets have the Baire property+the union of less than ω2 many meager sets is meager”, answering two well-known open questions of Woodin and Judah, respectively.

Strong Measure Zero and Strongly Meager Sets

Strong Measure Zero and Strongly Meager Sets

ADDITIVE PROPERTIES OF SETS OF REAL NUMBERS AND AN INFINITE GAME

Abstract If X has strong measure zero aid if Y is contained in an F σ, set of measure zero, then X + Y has measure zero (Proposition 9). If X is a measure zero set with property s 0 and Y is a Sierpinski set, then X + Y has property s 0 (Theorem 12). If X is a meager set with property s 0 and Y is a Lusin set, then X + Y has property s 0 (Theorem 17). An infinite game is introduced, motivated by additive properties of certain classes of sets of real numbers.

Partition reals and the consistency of t > add(R)

AbstractWe show that it is consistent with ZFC that the additivity number add(R) of the ideal of meager sets of the real line is strictly greater than the tower number t of the reals. MSC: 03E35, 54D20.

Strong measure zero and strongly meager sets

We consider conjectures made by Prikry and Galvin concerning strong measure zero and strongly meager sets of real numbers.

Ramsey sets, the Ramsey ideal, and other classes overR

AbstractWe improve results of Marczewski, Frankiewicz, Brown, and others comparing theω-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis ofω-ideals to include the completely Ramsey null sets relative to a Ramsey ultrafilter and obtain all 32 possible examples of sets in some ideals and not others, some under the assumption of MA, but most in ZFC alone. We also improve upon the known constructions of a Marczewski measure zero set which is not Ramsey by using a set theoretic hypothesis which is weaker than those used by other authors. We give several consistency proofs: one concerning the relative sizes of the covering numbers for the meager sets and the completely Ramsey null sets; another concerning the size of non; and a third concerning the size of add. We also study those classes of perfect sets which are bases for the class of always first category sets.

Closed measure zero sets

Bartoszynski, T. and S. Shelah, Closed measure zero sets, Annals of Pure and Applied Logic 58 (1992) 93–110. We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets.

RECURSIVE BAIRE CLASSIFICATION AND SPEEDABLE FUNCTIONS

AbstractUsing recursive variants of Baire notions of nowhere dense and meagre sets we study the topological size of speedable and infinitely often speedable functions in a machine‐independent framework. We show that the set of speedable functions is not “small” whereas the set of infinitely often speedable functions is “large”. In this way we offer partial answers to a question in [4].

$U$-Meager Sets when the Cofinality and the Coinitiality of $U$ are Uncountable

We prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set Xf = { I, . . , HI, where H is an infinite integer, cannot be expressed as a countable union of countably determined each of which is Umeager for some cut U with min {cf(U), ci(U)} ? co,. Also, every Borel, Z,1 or countably determined set C which is U-meager for every cut U is a countable union of Borel, Z1 or countably determined respectively, which are U-nowhere dense for every cut U. Further, the class of Borel U-meager for min{cf(U),ci(U)} ? wo1 coincides with the least family of containing internal U-meager and closed with respect to the operation of countable union and intersection. The same is true if the phrase Umeager sets is replaced by U-meager for every cut U' Introduction. In [KL] the authors introduced the notion of a U-meager and Unowhere dense set in the set Xf = {1,... , H}, where H is an infinite integer in a nonstandard model of the superstructure over the set of standard integers co. The set U is an initial segment of X closed with respect to addition, and is called a cut. Now, a set M is called U-nowhere dense if, for every interval I of length > U, M has a gap of length > U in I. A set is U-meager if it is a countable union of U-nowhere dense sets. The reader is referred to [KL] for details about the U-meager and U-nowhere dense sets. The purpose of this paper is to try to give some answers to some open questions from [KL], as well as to reveal some properties of U-meager for which the cofinality cf(U) and the coinitiality ci(U) are both uncountable. Here the cofinality of U is defined as the least K such that U has a cofinal subset of cardinality Kc, and the coinitiality of U is the least K such that -$ U has a coinitial subset of cardinality K. The Borel hierarchy of subsets of a hyperfinite set is defined in the usual manner, i.e., internal are called H1 or Z? and, by induction, for every a < co1, a set is HO (Z) if it is a countable intersection (union) of Z1 (HO) for fi < a. A set is Borel if it belongs to the union of the above defined classes. Received May 3, 1990; revised August 29, 1990. ? 1991, Association for Symbolic Logic 0022-4812/91/5603-001 5/$01.90

Relativized topological size of sets of partial recursive functions

In [1], a recursive topology on the set of unary partial recursive functions was introduced and recursive variants of Baire topological notions of nowhere dense and meagre sets were defined. These tools were used to measure the size of some classes of partial recursive (p.r.) functions. Thus, for example, it was proved that measured sets or complexity classes are recursively meagre in contrast with the sets of all p.r. functions or recursive functions, which are sets of recursively second Baire category. In this paper we measure the size of sets of p.r. functions using the above Baire notions relativized to the topological spaces induced by these sets. In this way we strengthen, in a uniform way, most results of [4, 5, 6, 3, 2], and we also obtain new results. For many sets of p.r. functions, strong differences between “local” and “global” topological size are established.

U-meager sets when the cofinality and the coinitiality of U are uncountable

AbtractWe prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set , where H is an infinite integer, cannot be expressed as a countable union of countably determined sets each of which is U-meager for some cut U with min{cf(U), ci(U)} ≥ ω1. Also, every Borel, or countably determined set C which is U-meager for every cut U is a countable union of Borel, or countably determined sets respectively, which are U-nowhere dense for every cut U. Further, the class of Borel U-meager sets for min{cf(U),ci(U)} ≥ ω1 coincides with the least family of sets containing internal U-meager sets and closed with respect to the operation of countable union and intersection. The same is true if the phrase “U-meager sets” is replaced by “U-meager for every cut U.”

Products of Perfectly Meagre Sets

Products of Perfectly Meagre Sets

Nonnormality or hereditary paracompactness of some spaces of real functions

The topology we give a set F of real functions is determined not by ε-balls, with ε>0, but by e-balls, with e ϵ F strictly positive. For F we will consider C( X), the continous real functions on a space X, or L and Ba, the real functions on R which are Lebesgue, or Baire measurable; in this case we are interested in the quotient spaces L/∼, and Ba/∼, with f∼ g if f agrees with g except on a null set, or on a meager set. We first show that C R is not normal, and then generalize this and prove nonnormality for nondiscrete, first countable or locally compact, nonpseudocompact X. We find a nonpseudocompact space S without isolated points such that C( S) is hereditarily paracompact under CH, the Continuum Hypothesis. We next show that L/∼ is hereditarily paracompact under MA, Martin's Axiom, but that Ba/∽ is not hereditarily normal. Also, L/∼ and Ba/∼ are proved nonhomeomorphic without using MA. We also consider a topology on C( X) induced by open subsets of X× R .

Meager sets on the hyperfinite time line

In this paper we study notions of a “meager subset” of a hyper-finite set. We work within an ω-saturated nonstandard universe and fix a hyperfinite natural number Є *N∖N. We shall consider subsets of the set = {1, 2, …,H}.By analogy with the meager subsets of the real interval [0, 1], a notion of meager subset of should have the following properties.1. Finite sets, countable unions of meager sets, subsets of meager sets, and translates of meager sets should be meager.2. The Baire Category Theorem should hold; that is, should not be meager.3. The internal analogue of the Cantor set should be meager.4. The notion of a meager set should be with respect to a natural topology on .5. There should exist meager subsets of of Loeb measure one.6. Sierpiński and Lusin sets should have hyperfinite counterparts with properties similar to the classical case.Property 5 is desirable so that, as with Lebesgue measure and Baire category on [0, 1], the topological and measure-theoretic notions of “large” and “small” sets are incomparable.Property 6, while not as necessary as the other five, is desirable because of the strong interplay between measure and category in the classical results about Lusin and Sierpinski sets. Our objective is to find notions of meager set which have a relationship to Loeb measure similar to the classical relationship between meager sets and Lebesgue measure.