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

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 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 Dedekindfinite 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 ZF that every non-discrete compact metrizable space contains an infinite compact scattered subspace are also included.

$k$-spaces, sequential spaces and related topics in the absence of the axiom of choice

“…Let us recall that a topological space X is called scattered (or, equivalently, dispersed ) if no non-empty subspace of X is dense-in-itself. The newest results on scattered spaces in ZF are included in [21].…”