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

TL;DR

This paper explores properties of various topological spaces within ZF set theory, establishing new results and principles that do not rely on the axiom of choice, and clarifying the relationships between these spaces.

Contribution

It introduces new choice principles and proves several theorems about topological spaces like $k$-spaces and metrizable spaces without using the axiom of choice.

Findings

Every Loeb, $T_3$-space with a countable union of finite base is second-countable and metrizable.

Every $G_{ ext{delta}}$-subspace of a second-countable, Cantor completely metrizable space is also Cantor completely metrizable.

The statement that every second-countable metrizable space is a very $k$-space is equivalent to the axiom of countable choice for $ extbf{R}$.

Abstract

In the absence of the axiom of choice, new results concerning sequential, Fr\'echet-Urysohn, $k$-spaces, very $k$-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other theorems, it is proved in $\mathbf{ZF}$ that every Loeb, $T_3$-space having a base expressible as a countable union of finite sets is a metrizable second-countable space whose every $F_{\sigma}$-subspace is separable; moreover, every $G_{\delta}$-subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. It is also noticed that Arkhangel'skii's statement that every very $k$-space is Fr\'echet-Urysohn is unprovable in $\mathbf{ZF}$ but it holds in $\mathbf{ZF}$ that every first-countable, regular very $k$-space whose family of all non-empty compact sets has a choice function is Fr\'echet-Urysohn. That every second-countable metrizable space is a very $k$-space is equivalent to the axiom of countable choice for $\mathbb{R}$.