A note on the Borel types of some small sets

2018 ◽  
Vol 25 (3) ◽  
pp. 419-425
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Real Line ◽  
The Real ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

AbstractThe Borel types of some classical small subsets of the real line are considered. In particular, under Martin’s axiom it is shown that there are at least {{\mathbf{c}}^{+}} pairwise incomparable Borel types of generalized Luzin sets (resp. of generalized Sierpiński sets), where {{\mathbf{c}}} stands for the cardinality of the continuum.

On measurability of algebraic sums of small sets

2008 ◽  
Vol 45 (3) ◽  
pp. 433-442
Author(s):  
Alexander Kharazishvili ◽  
Aleks Kirtadze
Keyword(s):  
Real Line ◽  
Measure Zero ◽  
The Real ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Nonmeasurable Set ◽  
Universal Measure

It is shown that, under Martin’s Axiom, the algebraic sum of two universal measure zero subsets of the real line can be an absolutely nonmeasurable set. Some related questions concerning measurability of algebraic sums of small sets are also discussed.

Uncountable Discrete Sets in Extensions and Metrizability

1982 ◽  
Vol 25 (4) ◽  
pp. 472-477 ◽  
Author(s):  
Murray Bell ◽  
John Ginsburg
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Continuum Hypothesis ◽  
Vietoris Topology ◽  
Discrete Subset ◽  
Similar Argument ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

AbstractIf X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.

Martin's axiom and Hausdorif measures

1974 ◽  
Vol 75 (2) ◽  
pp. 193-197 ◽  
Author(s):  
A. J. Ostaszewski
Keyword(s):  
Metric Space ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
The Continuum

AbstractA theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

The Character of Certain Closed Sets

1984 ◽  
Vol 36 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Mary Anne Swardson
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countably Compact ◽  
Countable Character ◽  
Image Position ◽  
The Continuum

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonalof a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

scholarly journals On generalizations of projectivity for modules over Dedekind domains

1981 ◽  
Vol 31 (2) ◽  
pp. 207-216 ◽  
Author(s):  
Jutta Hausen
Keyword(s):  
Exact Sequence ◽  
Structure Theorem ◽  
Continuum Hypothesis ◽  
Projective Modules ◽  
Generating Set ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Dedekind Domains ◽  
The Continuum

AbstractA module M over a ring R is κ-projective, κ a cardinal, if M is projective relative to all exact sequence of R-modules 0 → A → B → C → 0 such that C has a generating set of cardinality less than κ. A structure theorem for κ-projective modules over Dedekind domains is proven, and the κ-projectivity of M is related to properties of ExtR (M, ⊕ R). Using results of S. Chase, S. Shelah and P. Eklof, the existence of non-projective и1-projective modules is shown to undecidable, while both the Continuum Hypothesis and its denial (Plus Martin's Axiom) imply the existence of a reduced И0-projective Z-module which is not free.

An existence theorem for a special ultrafilter when 𝔡 = 𝔠

1993 ◽  
Vol 58 (4) ◽  
pp. 1359-1364
Author(s):  
James J. Moloney
Keyword(s):  
Lower Bound ◽  
Prime Ideal ◽  
Existence Theorem ◽  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Image Position ◽  
The Continuum ◽  
Convergent Sequences

For an ultrafilter , consider the ultrapower NN/. 〈an〉/ is in the top sky of NN/ if there exists a sequence 〈bn〉 ∈ NN such thatandIn [M2] we showed, assuming the Continuum Hypothesis, that there are exactly 10 c/p's (where c is the ring of real convergent sequences and p is a prime ideal of c). To get the lower bound we showed that there will be at least 10 c/p's in any model of ZFC where there exist both of the following kinds of ultrafilter:(i) nonprincipal P-points,(ii) non-P-points such that when the top sky is removed from NN/, the remaining model has countable cofinality.In [M2] we showed that the Continuum Hypothesis implies the existence of the ultrafilter in (ii). In this paper we show that its existence is implied by an axiom weaker than the Continuum Hypothesis, in fact weaker than Martin's Axiom, namely,(*) If is a subset of NN such that for any f: N → N there exists g ∈ such that g(n) > f(n) for all n, then ∣∣ = .

Remarks on Martin's Axiom and the Continuum Hypothesis

1991 ◽  
Vol 43 (4) ◽  
pp. 832-851 ◽  
Author(s):  
Stevo Todorcevic
Keyword(s):  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Image Position ◽  
The Continuum

Martin's axiom and the Continuum Hypothesis are studied here using the notion of accc partitioni.e., a partition of the formwhereK0has the following properties:(a)K0contains subsets of its elements as well as all singletons ofX.(b) Every uncountable subset of K0contains two elements whose union is inK0.

scholarly journals Some Consequences of Martin’s Axiom and the Negation of the Continuum Hypothesis

1973 ◽  
Vol 49 ◽  
pp. 117-125 ◽  
Author(s):  
Juichi Shinoda
Keyword(s):  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

W. Sierpisnki [3] demonstrated 82 propositions, called C1-C82, with the aid of the continuum hypothesis. D. A. Martin and R. M. Solovay remarked in [2] that 48 of these propositions followed from Martin’s axiom (MA), 23 were refuted by and three were independent of But the relation of the remaining eight propositions to has been unsettled.

scholarly journals Another proof of Hurewicz theorem

2011 ◽  
Vol 49 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Miroslav Repický
Keyword(s):  
Elementary Proof ◽  
Polish Space ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Isolated Points ◽  
The Continuum

ABSTRACT . A Hurewicz theorem says that every coanalytic non-Gδ set C in a Polish space contains a countable set Q without isolated points such that Q̅ ∩ C = Q. We present another elementary proof of this theorem and generalize it for k-Suslin sets. As a consequence, under Martin’s Axiom, we obtain a characterization of ∑12 sets that are the unions of less than the continuum closed sets.

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