Cardinal Functions, Bornologies and Strong Whitney convergence

TL;DR

This paper investigates the cardinal invariants of function spaces of continuous functions under strong Whitney and Whitney convergence topologies, providing new insights and simplified proofs of existing results.

Contribution

It introduces new results on cardinal invariants for these topologies and compares their relationships, simplifying some previous proofs in the literature.

Findings

Cardinal invariants of $C(X)$ under strong Whitney convergence are characterized.

Relationships between invariants of strong Whitney and strong uniform convergence are established.

Simplified proofs of existing results on function space topologies are provided.

Abstract

Let $C(X)$ be the set of all real valued continuous functions on a metric space $(X,d)$. Caserta introduced the topology of strong Whitney convergence on bornology for $C(X)$ in [A. Caserta, Strong Whitney convergence, Filomat, 2012], which is a generalization of the topology of strong uniform convergence on bornology introduced by Beer-Levi in [Beer-Levi, Strong uniform continuity, J. Math. Anal. Appl., 2009]. The purpose of this paper is to study various cardinal invariants of the function space $C(X)$ endowed with the topologies of strong Whitney and Whitney convergence on bornology. In the process, we present simpler proofs of a number of results from the literature. In the end, relationships between cardinal invariants of strong Whitney convergence and strong uniform convergence on $C(X)$ have also been studied.