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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

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 , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product [ 0 , 1 ] is compact. (iv) In a Boolean algebra...

On subcompactness and countable subcompactness of metrizable spaces in ZF

Kyriakos Keremedis — 2022

Commentationes Mathematicae Universitatis Carolinae

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space 𝐗 = ( X , T ) is countably compact if and only if it is countably subcompact relative to T . (iii) For every metrizable space 𝐗 = ( X , T ) , the following are equivalent: (a) 𝐗 is compact; (b) for every open filter of 𝐗 , { F ¯ : F } ; (c) 𝐗 is subcompact relative to T . We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable...

On pseudocompactness and related notions in ZF

Kyriakos Keremedis — 2018

Commentationes Mathematicae Universitatis Carolinae

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.

Some versions of second countability of metric spaces in ZF and their role to compactness

Kyriakos Keremedis — 2018

Commentationes Mathematicae Universitatis Carolinae

In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every ε > 0 , every cover by open balls of radius ε has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of holds true. (iii) A countably compact metric space is separable if and only if it is second countable.

On the metric reflection of a pseudometric space in ZF

Horst HerrlichKyriakos Keremedis — 2015

Commentationes Mathematicae Universitatis Carolinae

We show: (i) The countable axiom of choice 𝐂𝐀𝐂 is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom 𝐂𝐌𝐂 is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice 𝐀𝐂 is equivalent to each one of the...

On BPI Restricted to Boolean Algebras of Size Continuum

Eric HallKyriakos Keremedis — 2013

Bulletin of the Polish Academy of Sciences. Mathematics

(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 algebra of...

Products, the Baire category theorem, and the axiom of dependent choice

Horst HerrlichKyriakos Keremedis — 1999

Commentationes Mathematicae Universitatis Carolinae

In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (i) The axiom of dependent choice. (ii) Products of compact Hausdorff spaces are Baire. (iii) Products of pseudocompact spaces are Baire. (iv) Products of countably compact, regular spaces are Baire. (v) Products of regular-closed spaces are Baire. (vi) Products of Čech-complete spaces are Baire. (vii) Products of pseudo-complete spaces are Baire.

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

Horst HerrlichKyriakos KeremedisEleftherios Tachtsis — 2011

Bulletin of the Polish Academy of Sciences. Mathematics

In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2 ( X ) , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

Countable Compact Scattered T₂ Spaces and Weak Forms of AC

Kyriakos KeremedisEvangelos FelouzisEleftherios Tachtsis — 2006

Bulletin of the Polish Academy of Sciences. Mathematics

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...

On preimages of ultrafilters in ZF

Horst HerrlichPaul HowardKyriakos Keremedis — 2016

Commentationes Mathematicae Universitatis Carolinae

We show that given infinite sets X , Y and a function f : X Y which is onto and n -to-one for some n , the preimage of any ultrafilter of Y under f extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms A and a finite-to-one onto function f : A ω such that for each free ultrafilter of ω its preimage under f does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact pseudometric...

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