Remarks on the Stone Spaces of
the Integers and the Reals without ${\bf AC}$

Horst Herrlich; Kyriakos Keremedis; Eleftherios Tachtsis

doi:10.4064/ba59-2-1

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Remarks on the Stone Spaces of the Integers and the Reals without ${\bf AC}$

Horst Herrlich ¹ ; Kyriakos Keremedis ² ; Eleftherios Tachtsis ³

¹ Feldhäuser Str. 69 28865 Lilienthal, Germany
² Department of Mathematics University of the Aegean Karlovassi, Samos 83200, Greece
³ Department of Statistics and Actuarial-Financial Mathematics University of the Aegean Karlovassi, Samos 83200, Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) no. 2, pp. 101-114.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

Résumé

In $\mathbf{ZF}$, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice $\mathbf{AC}$, we investigate the relationship between the Tychonoff product $\mathbf{2}^{\mathcal{P}(X)}$, where $\mathbf{2}$ is $2=\{0,1\}$ with the discrete topology, and the Stone space $S(X)$ of the Boolean algebra of all subsets of $X$, where $X=\omega,\mathbb{R}$. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

DOI : 10.4064/ba59-2-1

Keywords: mathbf zermelo fraenkel set theory minus axiom choice mathbf investigate relationship between tychonoff product mathbf mathcal where mathbf discrete topology stone space boolean algebra subsets nbsp where omega mathbb study possible placement well known topological statements which concern cited spaces hierarchy weak choice principles

Affiliations des auteurs :

Horst Herrlich ¹ ; Kyriakos Keremedis ² ; Eleftherios Tachtsis ³

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Horst Herrlich; Kyriakos Keremedis; Eleftherios Tachtsis. Remarks on the Stone Spaces of
the Integers and the Reals without ${\bf AC}$. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) no. 2, pp. 101-114. doi : 10.4064/ba59-2-1. http://geodesic.mathdoc.fr/articles/10.4064/ba59-2-1/

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