SEPARABILITY OF PATH SPACES UNDER THE OPEN-POINT AND BI-POINT-OPEN TOPOLOGIES

Abstract

In [3] (Anubha Jindal, R. A. McCoy, and S. Kundu. The open-point and bi-point-open topologies on C(X), Topology and its Applications, 187:62-74, 2015.), two new kinds of topologies called the open-point topology and the bi-point-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space $X$, have been introduced. In this article, we study the separability of the space P(X), of all continuous maps on [0, 1] into a Hausdorff space X, with the open-point and bi-point-open topologies. Our result also demonstrates, the claim made in [3], that both the domain as well as the codomain play significant roles in the construction of the open-point and bi-point-open topologies.