On iso-dense and scattered spaces in $\mathbf{ZF}$

TL;DR

This paper investigates properties of iso-dense and scattered spaces within ZF set theory, constructing models and proving equivalences related to compactness, separability, and the axiom of choice.

Contribution

It introduces a new permutation model, establishes a metrization theorem for quasi-metric spaces, and links properties of metric spaces to the countable axiom of choice.

Findings

A discrete weakly Dedekind-finite space can have the Cantor set as a remainder.

Certain properties of metric iso-dense spaces are equivalent to the countable axiom of choice.

The paper explores whether all non-discrete compact metrizable spaces contain infinite scattered subspaces in ZF.

Abstract

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties of iso-dense spaces are investigated. A new permutation model is constructed in which a discrete weakly Dedekind-finite space can have the Cantor set as a remainder. A metrization theorem for a class of quasi-metric spaces is deduced. The statement "every compact scattered metrizable space is separable" and several other statements about metric iso-dense spaces are shown to be equivalent to the countable axiom of choice for families of finite sets. Results concerning the problem of whether it is provable in $\mathbf{ZF}$ that every non-discrete compact metrizable space contains an infinite compact scattered subspace are also included.