Remarks on the Stone Spaces of the Integers and the Reals without AC

Horst Herrlich; Kyriakos Keremedis; Eleftherios Tachtsis

Displaying similar documents to “Remarks on the Stone Spaces of the Integers and the Reals without AC”

Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem

Kyriakos Keremedis (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of (X) can be extended to an ultrafilter. UF(X): (X) has a free ultrafilter. We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product ${[0, 1]}^{ℝ}$ is compact. (iv) In a...

On the set-theoretic strength of the n-compactness of generalized Cantor cubes

Paul Howard, Eleftherios Tachtsis (2016)

Fundamenta Mathematicae

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We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product $2^{X}$ , where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel...

On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall, Kyriakos Keremedis (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product $2^{(ω)}$ the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of $2^{(ω)}$ to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...

On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " $2^{ℝ}$ is countably compact" and " $2^{ℝ}$ is compact"

Axioms weaker then $R_{0}$

J. Guia (1984)

Matematički Vesnik

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Boolean algebras, splitting theorems, and $Δ_{2}^{0}$ sets

Michael Morley, Robert Soare (1975)

Fundamenta Mathematicae

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$\forall_{n}$ -theories of Boolean algebras

Jan Waszkiewicz (1974)

Colloquium Mathematicae

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The reaping and splitting numbers of nice ideals

Rafał Filipów (2014)

Colloquium Mathematicae

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We examine the splitting number (B) and the reaping number (B) of quotient Boolean algebras B = (ω)/ℐ where ℐ is an $F_{σ}$ ideal or an analytic P-ideal. For instance we prove that under Martin’s Axiom ((ω)/ℐ) = for all $F_{σ}$ ideals ℐ and for all analytic P-ideals ℐ with the BW property (and one cannot drop the BW assumption). On the other hand under Martin’s Axiom ((ω)/ℐ) = for all $F_{σ}$ ideals and all analytic P-ideals ℐ (in this case we do not need the BW property). We also provide applications...

On $K$ -Boolean Rings

W. B. Vasantha Kandasamy (1992)

Publications du Département de mathématiques (Lyon)

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An axiomatics of non-Desarguean geometry based on the half-plane as the primitive notion

A. Śniatycki

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CONTENTSIntroduction................................................................................................................................................. 5PART I1. Axioms of Boolean algebra................................................................................................................. 62. Half-planes and their axioms.............................................................................................................. 73. The line.......................................................................................................................................................

Generalised irredundance in graphs: Nordhaus-Gaddum bounds

Ernest J. Cockayne, Stephen Finbow (2004)

Discussiones Mathematicae Graph Theory

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For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s,S), q = q(s,S) and r = r(s,S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p,q,r) may be considered as a compound existence property of S-pns. The subset S is called an f-set of G if f = 1 for all s ∈ S and the class of f-sets of G is denoted by $Ω_{f} (G)$ . Only 64 Boolean functions f can produce different classes $Ω_{f} (G)$ , special cases...

On pseudocompactness and related notions in ZF

Kyriakos Keremedis (2018)

Commentationes Mathematicae Universitatis Carolinae

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We study in ZF and in the class of $T_{1}$ spaces the web of implications/ non-implications between the notions of pseudocompactness, light compactness, countable compactness and some of their ZFC equivalents.

Axiom $T_{D}$ and the Simmons sublocale theorem

Jorge Picado, Aleš Pultr (2019)

Commentationes Mathematicae Universitatis Carolinae

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More precisely, we are analyzing some of H. Simmons, S. B. Niefield and K. I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_{D}$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation between sublocales and subspaces. S. B. Niefield and K. I. Rosenthal just mention this axiom in a remark about Simmons’ result....

Hyperplanes in matroids and the axiom of choice

Marianne Morillon (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC $^{fin}$ , the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC $^{fin}$ in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?

The gap between I₃ and the wholeness axiom

Paul Corazza (2003)

Fundamenta Mathematicae

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∃κI₃(κ) is the assertion that there is an elementary embedding $i : V_{λ} \to V_{λ}$ with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance...

Orthomodular lattices that are horizontal sums of Boolean algebras

Ivan Chajda, Helmut Länger (2020)

Commentationes Mathematicae Universitatis Carolinae

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The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class $ℋ$ of horizontal sums of Boolean algebras, we establish an identity...

Laslett’s transform for the Boolean model in $ℝ^{d}$

Rostislav Černý (2006)

Kybernetika

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Consider a stationary Boolean model $X$ with convex grains in $ℝ^{d}$ and let any exposed lower tangent point of $X$ be shifted towards the hyperplane $N_{0} = {x \in ℝ^{d} : x_{1} = 0}$ by the length of the part of the segment between the point and its projection onto the $N_{0}$ covered by $X$ . The resulting point process in the halfspace (the Laslett’s transform of $X$ ) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie...