Finite support iteration and strong measure zero sets

1990 ◽  
Vol 55 (2) ◽  
pp. 674-677
Author(s):  
Janusz Pawlikowski
Keyword(s):  
Real Line ◽  
Measure Zero ◽  
Finite Support ◽  
The Real ◽  
Zero Sets ◽  
Strong Measure ◽  
Borel Conjecture ◽  
Strong Measure Zero ◽  
Finite Support Iteration

AbstractAny finite support iteration of posets with precalibre ℵ1 which has the length of cofinahty greater than ω1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ1 strong measure zero sets.

scholarly journals Remarks on small sets on the real line Remarks on small sets on the real line

2009 ◽  
Vol 42 (1) ◽  
pp. 73-80
Author(s):  
Małgorzata Filipczak ◽  
Elžbieta Wagner-Bojakowska
Keyword(s):  
Real Line ◽  
Measure Zero ◽  
The Real ◽  
Strong Measure ◽  
Microscopic Set ◽  
Strong Measure Zero

Abstract We consider two kinds of small subsets of the real line: the sets of strong measure zero and the microscopic sets. There are investigated the properties of these sets. The example of a microscopic set, which is not a set of strong measure zero, is given.

Strong measure zero sets and rapid filters

1988 ◽  
Vol 53 (2) ◽  
pp. 393-402 ◽  
Author(s):  
Jaime I. Ihoda
Keyword(s):  
Measure Zero ◽  
Zero Sets ◽  
Strong Measure ◽  
Borel Conjecture ◽  
Strong Measure Zero

AbstractWe prove that cons(ZF) implies cons(ZF + Borel conjecture + there exists a Ramsey ultrafilter). We also prove some results on strong measure zero sets from the existence of generalized Luzin sets. We study the relationships between strong measure zero sets and rapid filters on ω.

Finite powers of strong measure zero sets

1999 ◽  
Vol 64 (3) ◽  
pp. 1295-1306 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Measure Zero ◽  
General Context ◽  
Partition Relation ◽  
Compact Metric Space ◽  
Totally Bounded ◽  
Zero Sets ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

The Baire category theorem and cardinals of countable cofinality

1982 ◽  
Vol 47 (2) ◽  
pp. 275-288 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Real Line ◽  
Measure Zero ◽  
Baire Category ◽  
The Other ◽  
Baire Category Theorem ◽  
The Real ◽  
Zero Sets ◽  
Category Theorem ◽  
Dense Sets ◽  
The Ideal

AbstractLet κB be the least cardinal for which the Baire category theorem fails for the real line R. Thus κB is the least κ such that the real line can be covered by κ many nowhere dense sets. It is shown that κB cannot have countable cofinality. On the other hand it is consistent that the corresponding cardinal for 2ω1 be ℵω. Similar questions are considered for the ideal of measure zero sets, other ω1, saturated ideals, and the ideal of zero-dimensional subsets of Rω1.

A characterization of strong measure zero sets

1996 ◽  
Vol 93 (1) ◽  
pp. 171-183 ◽  
Author(s):  
Janusz Pawlikowski
Keyword(s):  
Measure Zero ◽  
Zero Sets ◽  
Strong Measure ◽  
Strong Measure Zero

The algebraic sum of sets of real numbers with strong measure zero sets

1998 ◽  
Vol 63 (1) ◽  
pp. 301-324 ◽  
Author(s):  
Andrej Nowik ◽  
Marion Scheepers ◽  
Tomasz Weiss
Keyword(s):  
Measure Zero ◽  
Real Numbers ◽  
Covering Property ◽  
Zero Sets ◽  
First Category ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractWe prove the following theorems:(1) IfXhas strong measure zero and ifYhas strong first category, then their algebraic sum has property S0.(2) IfXhas Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set.(3) IfXhas strong measure zero and Hurewicz's covering property then its algebraic sum with any set inis a set in. (is included in the class of sets always of first category, and includes the class of strong first category sets.)These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the γ-property and of a first category set is a first category set, and Bartoszyfński and Judah's characterization of-sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts.

Strong Measure Zero Sets and Rapid Filters

1988 ◽  
Vol 53 (2) ◽  
pp. 393 ◽  
Author(s):  
Jaime I. Ihoda
Keyword(s):  
Measure Zero ◽  
Zero Sets ◽  
Strong Measure ◽  
Strong Measure Zero

scholarly journals Strong measure zero sets without Cohen reals

1993 ◽  
Vol 58 (4) ◽  
pp. 1323-1341 ◽  
Author(s):  
Martin Goldstern ◽  
Haim Judah ◽  
Saharon Shelah
Keyword(s):  
Measure Zero ◽  
Zero Set ◽  
Zero Sets ◽  
Cohen Real ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractIf ZFC is consistent, then each of the following is consistent with :(1) X ⊆ ℝ is of strong measure zero iff ∣X∣ ≤ ℵ1 + there is a generalized Sierpinski set.(2) The union of ℵ many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size ℵ2 + there is no Cohen real over L.

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