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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2016 | 234 | 3 | 241-252

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

EN

### Abstrakty

EN
We investigate, in set theory without the Axiom of Choice 𝖠𝖢, 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 𝖹𝖥 (Zermelo-Fraenkel set theory minus 𝖠𝖢), equivalent to the Boolean Prime Ideal Theorem 𝖡𝖯𝖨, whereas
(2) Q(2) is strictly weaker than 𝖡𝖯𝖨 in 𝖹𝖥𝖠 set theory (Zermelo-Fraenkel set theory with the Axiom of Extensionality weakened in order to allow atoms).
This settles the open problem in Tachtsis (2012) on the relation of Q(n), n = 2,3,4,5, to 𝖡𝖯𝖨.

241-252

wydano
2016

### Twórcy

autor
• Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U.S.A.
autor
• Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece