On Loeb and sequential spaces in $\mathbf{ZF}$

Abstract

A topological space is called Loeb if the collection of all its non-empty closed sets has a choice function. In this article, in the absence of the axiom of choice, connections between Loeb and sequential spaces are investigated. Among other results, it is proved in $\mathbf{ZF}$ that if $\mathbf{X}$ is a Cantor completely metrizable second-countable space, then $\mathbf{X}^{\omega}$ is Loeb. If a sequential, sequentially locally compact space $\mathbf{X}$ has the property that every infinitely countable family of non-empty closed subsets of $\mathbf{X}$ has a choice function, then the Cartesian product $\mathbf{X}\times\mathbf{Y}$ of $\mathbf{X}$ with any sequential space $\mathbf{Y}$ is sequential. In consequence, it holds true in $\mathbf{ZF}$ that the Cartesian product of a sequential locally countably compact space with any sequential space is sequential. If $\mathbb{R}$ is sequential, then every second-countable compact Hausdorff space is sequential. It is also proved that, in some models of $\mathbf{ZF}$, a countable product of Cantor completely metrizable second-countable spaces can fail to be Loeb and it is independent of $\mathbf{ZF}$ that every sequential subspace of $% \mathbb{R}$ is Loeb. Some other sentences are shown to be independent of $% \mathbf{ZF}$. Several open problems are posed, among them, the following question: is $\mathbb{R}^{\omega}$ sequential if $\mathbb{R}$ is sequential?