Cover topologies, subspaces, and quotients for some spaces of vector-valued functions

Abstract

‎Let $X$ be a completely regular Hausdorff space‎, ‎and let $\mathcal{D}$ be a‎ ‎cover of $X$ by $C_{b}$-embedded sets‎. ‎Let $\pi‎: ‎\mathcal{E} \rightarrow X$‎ ‎be a bundle of Banach spaces (algebras)‎, ‎and let $\Gamma(\pi)$ be the‎ ‎section space of the bundle $\pi‎.‎$ Denote by $\Gamma _{b}(\pi‎,‎\mathcal{D})$‎ ‎the subspace of $\Gamma (\pi )$ consisting of sections which are bounded on‎ ‎each $D \in \mathcal{D}$. We construct a bundle $\rho ^{\prime }: \mathcal{F}‎^{\prime}\rightarrow \beta X$ such that $\Gamma _{b}(\pi‎ ,‎ \mathcal{D}) ‎$ is topologically and algebraically isomorphic to $\Gamma(\rho^\prime‎‎)‎$, ‎and use this to study the subspaces (ideals) and quotients resulting‎ ‎from endowing $\Gamma _{b}(\pi‎,‎\mathcal{D})$ with the cover topology‎ ‎determined by $\mathcal{D}$‎.