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On pseudocompactness and light compactness of metric spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2018

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Vol. 66, no. 2

99--113

EN

In the realm of metric spaces we show, in the Zermelo-Fraenkel set theory ZF, that: (a) A metric space X = (X, d) is countably compact iff it is pseudocompact. (b) Given a metric space X = (X, d); the following statements are equivalent: (i) X is lightly compact (every locally finite family of open sets is finite). (ii) Every locally finite family of subsets of X is finite. (iii) Every locally finite family of closed subsets of X is finite. (iv) Every pairwise disjoint, locally finite family of subsets of X is finite. (v) Every pairwise disjoint, locally finite family of closed subsets of X is finite. (vi) Every locally finite, pairwise disjoint family of open subsets of X is finite. (vii) Every locally finite open cover of X has a finite subcover. (c) For every infinite set X, the powerset P(X) of X has a countably infinite subset iff every countably compact metric space is lightly compact.

2

Uniform continuity and normality of metric spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2017

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Vol. 65, no. 2

113--124

EN

Let X = (X, d) and Y = (Y, ρ) be two metric spaces. (a) We show in ZF that: (i) If X is separable and f: X → Y is a continuous function then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0. But it is relatively consistent with ZF that there exist metric spaces X, Y and a continuous, nonuniformly continuous function f : X → Y such that for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0. (ii) If S is a dense subset of X, Y is Cantor complete and f : S → Y a uniformly continuous function, then there is a unique uniformly continuous function F : X → Y extending f. But it is relatively consistent with ZF that there exist a metric space X, a complete metric space Y, a dense subset S of X and a uniformly continuous function f : S → Y that does not extend to a uniformly continuous function on X. (iii) X is complete iff for any Cauchy sequences (xn)n∈N and (yn)n∈N in X, if [wzór] then d({xn : n ∈ N},{yn : n ∈ N}) > 0. (b) We show in ZF+CAC that if f : X → Y is a continuous function, then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0.

3

Some Notions of Separability of Metric Spaces in ZF and Their Relation to Compactness

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 2/3

119--136

EN

In the realm of metric spaces we show in ZF that: (1) Quasi separability (a metric space X = (X, d) is quasi separable iff X has a dense subset which is expressible as a countable union of finite sets) is the weakest property under which a limit point compact metric space is compact. (2) ω-quasi separability (a metric space X = (X, d) is ω-quasi separable iff X has a dense subset which is expressible as a countable union of countable sets) is a property under which a countably compact metric space is compact. (3) The statement “Every totally bounded metric space is separable” does not imply the countable choice axiom CAC.

4

On Sequential Compactness and Related Notions of Compactness of Metric Spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 1

29--46

EN

We show that: (i) If every sequentially compact metric space is countably compact then for every infinite set X, [X]<ω is Dedekind-infinite. In particular, every infinite subset of R is Dedekind-infinite. (ii) Every sequentially compact metric space is compact iff every sequentially compact metric space is separable. In addition, if every sequentially compact metric space is compact then: every infinite set is Dedekind-infinite, the product of a countable family of compact metric spaces is compact, and every compact metric space is separable. (iii) The axiom of countable choice implies that every sequentially bounded metric space is totally bounded and separable, every sequentially compact metric space is compact, and every uncountable sequentially compact, metric space has size |R|. (iv) If every sequentially bounded metric space is totally bounded then every infinite set is Dedekind-infinite. (v) The statement: “Every sequentially bounded metric space is bounded” implies the axiom of countable choice restricted to the real line. (vi) The statement: “For every compact metric space X either |X| ≤ |R|, or |R| ≤ |X|” implies the axiom of countable choice restricted to families of finite sets. (vii) It is consistent with ZF that there exists a sequentially bounded metric space whose completion is not sequentially bounded. (viii) The notion of sequential boundedness of metric spaces is countably productive.

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On BPI Restricted to Boolean Algebras of Size Continuum

Hall E., Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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Vol. 61, no 1

9--21

EN

(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 2P(ω) the restriction B|Y={Y∩B:B∈B} of the standard clopen base B of 2P(ω) to Y has size ≤|P(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of P(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 size ≤|R| has an ultrafilter.

6

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

Herrlich H., Keremedis K., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 2

101--114

EN

In ZF, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2P(X), where 2 is 2 = f0; 1g 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.

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On the compactness and countable compactness of 2R in ZF

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 4

293-302

EN

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

8

Countable compact scattered T2 spaces and weak forms of AC

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 1

75-84

EN

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 is scattered. (4) It is not provable in ZF+¬AC that there exists a countable compact T2 space which is dense-in-itself.

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Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2005

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Vol. 53, no 4

349-359

EN

Let X be an infinite set, and P(X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of P(X) can be extended to an ultrafilter. UF(X): P(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 2R, where 2 is the discrete space {0, 1}, is compact. (iii) The Tychonoff product [0, 1] R is compact. (iv) In a Boolean algebra of size ≤ |R| every filter can be extended to an ultrafilter. We will also show that in ZF, UF(R) does not imply BPI(R). Hence, BPI(R) is strictly stronger than UF(R). We do not know if UF(ω) implies BPI(ω) in ZF. Furthermore, we will prove that the axiom of choice for sets of subsets of R does not imply BPI(R) and, in addition, the axiom of choice for well orderable sets of non-empty sets does not imply BPI(ω).