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1

Preserving Dependent Choice

Karagila Asaf

Bulletin of the Polish Academy of Sciences. Mathematics

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2019

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Vol. 67, no. 1

19--29

EN

We prove some general theorems for preserving Dependent Choice when taking symmetric extensions, some of which are unwritten folklore results. We apply these to various constructions to obtain various simple consistency proofs.

2

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.

3

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.

4

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.

5

On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF

Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2015

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Vol. 63, no. 1

1--10

EN

In ZF (i.e. Zermelo–Fraenkel set theory without the Axiom of Choice AC), we investigate the relationship between UF(ω) (there exists a free ultrafilter on ω) and the statements ‶there exists a free ultrafilter on every Russell-set″ and ‶there exists a Russell-set A and a free ultrafilter F on A″. We establish the following results: 1. UF(ω) implies that there exists a free ultrafilter on every Russell-set. The implication is not reversible in ZF. 2. The statement there exists a free ultrafilter on every Russell-set″ is not provable in ZF. 3. If there exists a Russell-set A and a free ultrafilter on A, then UF(ω) holds. The implication is not reversible in ZF. 4. If there exists a Russell-set A and a free ultrafilter on A, then there exists a free ultrafilter on every Russell-set. We also observe the following: (a) The statements BPI(ω) (every proper filter on ω can be extended to an ultrafilter on ω) and ‶there exists a Russell-set A and a free ultrafilter F on A″ are independent of each other in ZF. (b) The statement ‶there exists a Russell-set and there exists a free ultrafilter on every Russell-set″ is, in ZF, equivalent to ‶there exists a Russell-set A and a free ultrafilter F on A″. Thus, ‶there exists a Russell-set and there exists a free ultrafilter on every Russell-set″ is also relatively consistent with ZF.

6

On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

Herrlich H., Howard P., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2015

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Vol. 63, no. 2

89--112

EN

We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice (AC), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y→X.

7

Weak homomorphisms between functorial algebra

Schneider F. M.

Demonstratio Mathematica

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2011

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Vol. 44, nr 4

801-818

EN

In universal algebra, homomorphisms are usually considered between algebras of the same similarity type. Different from that, the notion of a weak homomorphism, as introduced by E. Marczewski in 1961, does not depend on a signature, but only on the clones of term operations generated by the examined algebras. We generalize this idea by defining weak homomorphisms between F1 - and F2-algebras, where F1 and F2 denote not necessarily equal endofunctors of the category of sets. The aim is to show that, in many respects, weak homomorphisms behave very similarly to proper homomorphisms-without restricting the scope of considerations by the necessity of a common type. For instance, concerning a set F of Set -endofunctors that weakly preserve kernels, the class of all algebras of types from F equipped with the class of all weak homomorphisms between these algebras forms a category which admits a canonical factorization structure for morphisms. Furthermore, we treat two product constructions from which the notion of a weak homomorphism naturally arises.

8

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.

9

On the set-theoretic strength of countable compactness of the Tychonoff product 2R

Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2010

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Vol. 58, no 2

91-107

EN

We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets (ACWO) does not imply "the Tychonoff product 2R, where 2 is the discrete space {0,1}, is countably compact" in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets of reals imply 2R is countably compact in ZF? 2. Assuming the Countable Axiom of Multiple Choice (CMC), the statements "every infinite subset of 2R has an accumulation point", "every countably infinite subset of 2R has an accumulation point", "2R is countably compact", and UF(ω) = "there is a free ultrafilter on ω" are pairwise equivalent. 3. The statements "for every infinite set X, every countably infinite subset of 2X has an accumulation point", "every countably infinite subset of 2R has an accumulation point", and UF(ω) are, in ZF, pairwise equivalent. Hence, in ZF, the statement "2R is countably compact "implies UF(ω). 4. The statement "every infinite subset of 2R has an accumulation point" implies "every countable family of 2-element subsets of the powerset Ρ(R) of R has a choice function". 5. The Countable Axiom of Choice restricted to non-empty finite sets, (CACfin), is, in ZF, strictly weaker than the statement "for every infinite set X, 2X is countably compact".

10

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".

11

Spanning graphs and the Axiom of Choice

Delhommi Ch., Morillon M.

Reports on Mathematical Logic

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2006

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Nr 40

165-180

EN

We show in set-theory ZF that the axiom of choice is equivalent to the statement every bipartite connected graph has a spanning sub-graph omitting some complete finite bipartite graph Kn;n, and in particular it is equivalent to the fact that every connected graph has a spanning cycle-free graph (possibly non connected).

12

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.

13

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).

14

Some remarks about two definitions of continuity

Cichoń J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 1

43-47

EN

We formulate some variant of the axiom of choice which is neccessary and sufficient for the equivalence ofthe two definitions of continuity (the Cauchy and the Heine definition) for functions from the real line into any metric space. This result definitely solves the problem of the equivalence of these two classical definitions. We also slightly improve one of Sierpiński's results about global continuity of functions from the real line. We give a negative answer to the problem from Jaegermann's ciassical paper [2].

15

Argumenty heurystyczne w teorii mnogości

Wójtowicz K.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

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1998

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[Z] 34

61-82

16

On Scott consequence systems

Dimov G., Vakarelov D.

Fundamenta Informaticae

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1998

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Vol. 33, nr 1

43-70

EN

The notion of Scott consequence system (briefly, S-system) was introduced by D.Vakarelov in an analogy to a similar notion given by D. Scott. In part one of the paper we study the category Ssyst of all S-systems and all their morphisms. We show that the category DLat of all distributive lattices and all lattice homomorphisms is isomorphic to a reflective full subcategory of the category Ssyst. Extending the representation theory of D. Vakarelo for S-systems in P-systems, we develop an isomorphism theory for S-systems and for Tarski consequence systems. In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.