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

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 $\mathsf {ZF}$.