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1

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".

2

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.

3

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).

4

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].

5

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