A Banach-Dieudonn\'e theorem for the space of bounded continuous functions on a separable metric space with the strict topology

Let X be a separable metric space and let \beta be the strict topology on the space of bounded continuous functions on X, which has the space of \tau-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonne\'{e} type result for the space of bounded continuous functions equipped with \beta. As a consequence, this space is hypercomplete and a Pt\'{a}k space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.