The Bi-Compact-Open Topology on C(X)

TL;DR

This paper investigates the separation and countability properties of a new topology on C(X), formed by combining the compact-open and open-point topologies, to understand its topological characteristics.

Contribution

It introduces and analyzes the bi-compact-open topology on C(X), exploring its separation and countability properties, extending the understanding of set-open topologies.

Findings

Characterizes separation properties of the bi-compact-open topology.

Determines conditions for countability properties.

Provides insights into the structure of combined topologies on C(X).

Abstract

For the set C(X) of real-valued continuous functions on a Tychonoff space X, the compact-open topology on C(X) is a "set-open topology". This paper studies the separation and countability properties of the space C(X) having the topology given by the join of the compact-open topology and an "open-set topology" called the open-point topology, that was introduced in [6].