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

Volume 234 / 2016

Paul Howard, Eleftherios Tachtsis Fundamenta Mathematicae 234 (2016), 241-252 MSC: Primary 03E25; Secondary 03E35, 54B10, 54D30. DOI: 10.4064/fm961-1-2016 Published online: 4 July 2016

Abstract

We investigate, in set theory without the Axiom of Choice $\mathbf{\mathsf{AC}}$, 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 $\mathbf{\mathsf{ZF}}$ (Zermelo–Fraenkel set theory minus $\mathbf{\mathsf{AC}}$), equivalent to the Boolean Prime Ideal Theorem $\mathbf{\mathsf{BPI}}$, whereas

(2) $Q(2)$ is strictly weaker than $\mathbf{\mathsf{BPI}}$ in $\mathbf{\mathsf{ZFA}}$ 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 $\mathbf{\mathsf{BPI}}$.

Authors

  • Paul HowardDepartment of Mathematics
    Eastern Michigan University
    Ypsilanti, MI 48197, U.S.A.
    e-mail
  • Eleftherios TachtsisDepartment of Mathematics
    University of the Aegean
    Karlovassi 83200, Samos, Greece
    e-mail

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