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

TL;DR

This paper explores the properties of Loeb and sequential spaces within ZF set theory, establishing conditions under which certain spaces are Loeb or sequential, and demonstrating independence results and open problems related to these properties.

Contribution

It provides new results on the relationship between Loeb and sequential spaces in ZF, including conditions for product spaces to be sequential or Loeb, and shows independence of some properties from ZF.

Findings

If X is a Cantor completely metrizable second-countable space, then X^ω is Loeb.

The product of a sequential, locally countably compact space with any sequential space is sequential.

In some models of ZF, countable products of certain spaces can fail to be Loeb.

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?