Cardinal Properties of the Space of Quasicontinuous Functions under Topology of Uniform Convergence on Compact Subsets

TL;DR

This paper analyzes the cardinal properties of the space of quasicontinuous functions under uniform convergence on compact sets, revealing how these properties relate to the topology of the underlying space.

Contribution

It provides new insights into the relationships between tightness, Frechet-Urysohn property, and other cardinal invariants of quasicontinuous function spaces.

Findings

Tightness of $Q_{C}X$ aligns with the compact Lindelöf number of $X$ in Hausdorff spaces.

Countable tightness of $Q_{C}X$ when $X$ is second countable.

Conditions for tightness and Frechet-Urysohn property of $Q_{C}X$ related to $k$-covers and $\sigma$-compactness.

Abstract

In this paper, we investigate various cardinal properties of the space $Q_{C}X$ of all real-valued quasicontinuous functions on the topological space $X$, under the topology of uniform convergence on compact subsets. It begins by examining the relationship between tightness and other properties in the context of the space $X$, highlighting results such as the alignment of tightness $Q_{C}X$ with the compact Lindel\"of number of $X$ under Hausdorff conditions and the countable tightness of $Q_{C}X$ when $X$ is second countable. Further investigations reveal conditions for the tightness of $Q_{C}X$ relative to $k$-covers of $X$, as well as connections between density tightness, fan tightness, and other properties in Hausdorff spaces. Additionally, we discuss the implications of the Frechet-Urysohn property $Q_{C}X$ for open $k$-covers in Hausdorff spaces. We explore relationships between $Q_{C}X$'s tightness, the Frechet-Urysohn property, and the $\sigma$-compactness of locally compact Hausdorff spaces $X$. Furthermore, we examine the $k_{f}$-covering property and the existence of $k$-covers in the context of Whyburn spaces.