A study concerning splitting and jointly continuous topologies on $C(Y,Z)$

TL;DR

This paper introduces new family-open topologies on the space of continuous functions between two topological spaces, analyzing conditions for their splitting and joint continuity.

Contribution

It constructs and investigates ${ m extstyle extit{F}_n( au_n)}$-family-open topologies on $C(Y,Z)$, providing criteria for their splitting and joint continuity.

Findings

Established necessary and sufficient conditions for splitting topology.

Determined criteria for joint continuity of the topologies.

Proposed questions for future research in the area.

Abstract

Let $Y$ and $Z$ be two fixed topological spaces and $C(Y,Z)$ the set of all continuous maps from $Y$ into $Z$. We construct and study topologies on $C(Y,Z)$ that we call ${\cal F}_n(\tau_n)$-family-open topologies. Furthermore, we find necessary and sufficient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.