Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Juhász's topological generalization of Neumer's theorem may fail in $ \mathsf{ZF}$


Author: Eleftherios Tachtsis
Journal: Proc. Amer. Math. Soc. 148 (2020), 1295-1310
MSC (2010): Primary 03E25; Secondary 03E35, 54A35
DOI: https://doi.org/10.1090/proc/14794
Published electronically: October 28, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In set theory without the Axiom of Choice ( $ \mathsf {AC}$), we investigate the open problem of the deductive strength of Juhász's topological generalization of Neumer's Theorem from his paper On Neumer's Theorem [Proc. Amer. Math. Soc. 54 (1976), 453-454].

Among other results, we show that Juhász's Theorem is deducible from the Principle of Dependent Choices and (when restricted to the class of $ T_{1}$ spaces) implies the Axiom of Countable Multiple Choice, and hence implies van Douwen's Countable Choice Principle, but does not imply either the full van Douwen's Choice Principle or the axiom of choice for linearly ordered families of nonempty finite sets. Furthermore, we prove that Juhász's Theorem (for $ T_{1}$ spaces) implies each of the following principles: `` $ \aleph _{1}$ is a regular cardinal'', ``every infinite set is weakly Dedekind-infinite'', and ``every infinite linearly ordered set is Dedekind-infinite''. We also establish that Juhász's Theorem for $ T_{2}$ spaces is not provable in $ \mathsf {ZF}$.

In contrast to the above results, we show that Neumer's Theorem and Juhász's Theorem for compact $ T_{1}$ spaces are both provable in $ \mathsf {ZF}$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E25, 03E35, 54A35

Retrieve articles in all journals with MSC (2010): 03E25, 03E35, 54A35


Additional Information

Eleftherios Tachtsis
Affiliation: Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
Email: ltah@aegean.gr

DOI: https://doi.org/10.1090/proc/14794
Keywords: Axiom of Choice, weak axioms of choice, Juh\'asz's Theorem, Neumer's Theorem, Fraenkel--Mostowski permutation model of $\mathsf{ZFA}$, Jech--Sochor First Embedding Theorem, Pincus's transfer theorems.
Received by editor(s): February 14, 2019
Received by editor(s) in revised form: May 22, 2019, July 14, 2019, and July 21, 2019
Published electronically: October 28, 2019
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2019 American Mathematical Society