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About 4,092 papers for “Countable set”

Product set phenomena for countable groups

Michael Björklund, Alexander Fish · arXiv (Cornell University) · 2013

We develop in this paper general techniques to analyze local combinatorial structures in product sets of two subsets of a countable group which are "large" with respect to certain classes of (not necessarily invariant) means on the group. As applications of our methods, we extend and quantify a series of recent results by Jin, Bergelson-Furstenberg-Weiss, Beiglb\"ock-Bergelson-Fish, Griesmer and DiNasso-Lupini to general countable groups.

Cited by 0Open access

Martingales with a Countable Filtering Index Set

J.-P. Gabriel · The Annals of Probability · 1977

This paper is concerned with the almost everywhere convergence of martingales indexed by countable filtering sets. It is shown that the convergence is a consequence of the maximal inequality as it is in the classical case. It also contains some results about the law of large numbers when the index belongs to a sector and an optimal condition assuring the almost everywhere convergence of martingales in these sectors.

Cited by 19Open access

A sufficient condition for countable-set aposyndesis

Donald E. Bennett · Proceedings of the American Mathematical Society · 1972

In this paper a stronger form of aposyndesis is defined and continua with this property (strongly aposyndetic) are shown to be countable-set aposyndetic. Although every continuum which is a product of nondegenerate continua is countable-set aposyndetic, it is established that no product of nondegenerate continua is strongly aposyndetic.

Cited by 1

Countable dense homogeneity and the Cantor set

Hernández-Gutiérrez, Rodrigo · arXiv (Cornell University) · 2017

It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says that for any crowded Hausdorff space $X$ of countable $π$-weight, if ${}^ω{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set.

Cited by 0Open access

Power Set of Natural Numbers is Countable

Sonia Sharma · Zenodo (CERN European Organization for Nuclear Research) · 2022

This paper explains the Cardinality of the Power Set of Natural numbers. Set of Natural numbers is countable, in the same way the power set of Natural numbers is also countable as every subset of the Power set of Natural numbers is Countable. Prime numbers and Well ordering Principle play a very important role in proving this result. Since every Subset of Natural numbers is Countable, there exists a bijection between the Power Set of Natural numbers and a proper subset of Natural numbers.

Cited by 0Open access

A Sufficient Condition for Countable-Set Aposyndesis

Donald E. Bennett · Proceedings of the American Mathematical Society · 1972

In this paper a stronger form of aposyndesis is defined and continua with this property (strongly aposyndetic) are shown to be countable-set aposyndetic.Although every continuum which is a product of nondegenerate continua is countable-set aposyndetic, it is established that no product of nondegenerate continua is strongly aposyndetic.Throughout this paper a continuum is a compact connected metric space.Let M be a continuum.If N is a subcontinuum of M, the interior of N in M will be denoted by int N and the boundary of N in M by Bd N.2 A continuum M is aposyndetic at a point p if for each point q in M-{p} there is a subcontinuum N in the complement of {q} such that/>eint N [3].If the {q} above is replaced by a finite (countable closed) set then M is said to be finitely (countable-set) aposyndetic at p.A continuum is aposyndetic (finitely aposyndetic) (countable-set aposyndetic) if it has the given property at each point [7].This familiar property can be strengthened as follows.Definition.A decomposable continuum M is strongly aposyndetic provided that for any pair of proper subcontinua 77 and K such that M=HuK, each of 77 and K is aposyndetic.It is easily seen that each strongly aposyndetic continuum is aposyndetic and there are numerous elementary examples of aposyndetic continua which fail to be strongly aposyndetic.The following two theorems established sufficient conditions for subcontinua of a strongly aposyndetic continuum to be aposyndetic.The first theorem is an immediate consequence of the definition.Theorem 1.If A is a nonseparating subcontinuum of a strongly aposyndetic continuum M and int Aj 0, then A is aposyndetic.

Cited by 3Open access

Power Set of Natural Numbers is Countable

Sharma, Sonia · Zenodo (CERN European Organization for Nuclear Research) · 2022

This paper explains the Cardinality of the Power Set of Natural numbers. Set of Natural numbers is countable, in the same way the power set of Natural numbers is also countable as every subset of the Power set of Natural numbers is Countable. Prime numbers and Well ordering Principle play a very important role in proving this result. Since every Subset of Natural numbers is Countable, there exists a bijection between the Power Set of Natural numbers and a proper subset of Natural numbers.

Cited by 0Open access

Elementary extensions of countable models of set theory

John Hutchinson · Journal of Symbolic Logic · 1976

Abstract We prove the following extension of a result of Keisler and Morley. Suppose is a countable model of ZFC and c is an uncountable regular cardinal in . Then there exists an elementary extension of which fixes all ordinals below c , enlarges c , and either (i) contains or (ii) does not contain a least new ordinal. Related results are discussed.

Cited by 9

Spaces of countable free set number and PFA

Alan Dow, István Juhász · Proceedings of the American Mathematical Society · 2023

The main result of this paper is that, under PFA, for every <italic>regular</italic> space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis upper X right-parenthesis equals omega"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> ω </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">F(X) = \omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X EndAbsoluteValue less-than-or-equal-to w left-parenthesis upper X right-parenthesis Superscript omega"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi> ω </mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|X| \le w(X)^\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; in particular, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis upper X right-parenthesis less-than-or-equal-to German c"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≤ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">w(X) \le \mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X EndAbsoluteValue less-than-or-equal-to German c"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X| \le \mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis upper X right-parenthesis equals omega"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> ω </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">F(X) = \omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis upper X right-parenthesis equals German c"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">w(X) = \mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X EndAbsoluteValue equals 2 Superscript German c"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|X| = 2^\mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml

Cited by 0

ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET

Manoj Kumar Bose, Ajoy Mukharjee · 2010

Considering a countable number of topologies on a set X, we introduce the notion of (ℵ0)topological spaces as a generalization of the notions of both bitopological spaces and (ω)topological spaces, and study some of their properties.

Cited by 3

Set-Valued Fractal Approximation for Countable Data Sets

Kaur, Parneet, Lal, Rattan, Kumar, Ankit et al. · arXiv (Cornell University) · 2025

Fractal geometry deals mainly with irregularity and captures the complexity of a structure or phenomenon. In this article, we focus on the approximation of set-valued functions using modern machinery on the subject of fractal geometry. We first provide a construction of fractal functions for countable data sets and use these functions in the approximation and study of set-valued mappings. We also show the existence and uniqueness of an invariant Borel measure supported on the graph of a set-valued fractal function. In addition, we obtain some effective bounds on the dimensions of the constructed set-valued fractal functions.

Cited by 0Open access

Set-Valued Fractal Approximation for Countable Data Sets

Kaur, Parneet, Lal, Rattan, Kumar, Ankit et al. · ArXiv.org · 2025

Fractal geometry deals mainly with irregularity and captures the complexity of a structure or phenomenon. In this article, we focus on the approximation of set-valued functions using modern machinery on the subject of fractal geometry. We first provide a construction of fractal functions for countable data sets and use these functions in the approximation and study of set-valued mappings. We also show the existence and uniqueness of an invariant Borel measure supported on the graph of a set-valued fractal function. In addition, we obtain some effective bounds on the dimensions of the constructed set-valued fractal functions.

Cited by 0Open access