Frechet-Urysohn property of quasicontinuous functions

TL;DR

This paper investigates the conditions under which the space of quasicontinuous functions on a topological space exhibits the Frechet-Urysohn property, revealing that countability of the domain space is crucial under various set-theoretic assumptions.

Contribution

It establishes the equivalence between the Frechet-Urysohn property of quasicontinuous function spaces and the countability of the domain space, under different set-theoretic frameworks.

Findings

For open Whyburn spaces, Frechet-Urysohn property holds iff the space is countable under Suslin's Hypothesis.

In the class of first-countable regular spaces, the same equivalence is valid.

For metrizable spaces, the property is equivalent to the space being countable in ZFC.

Abstract

The aim of this paper is to study the Frechet-Urysohn property of the space $Q_p(X,\mathbb{R})$ of real-valued quasicontinuous functions, defined on a Hausdorff space $X$, endowed with the pointwise convergence topology. It is proved that under Suslin's Hypothesis, for an open Whyburn space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable. In particular, it is true in the class of first-countable regular spaces $X$. In ZFC, it is proved that for a metrizable space $X$, the space $Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable.