About the space of continuous functions with open domain
Edwar Alexis Ramírez Ardila
TL;DR
The work develops explicit metrizations for spaces of continuous functions with open domains by connecting the inverse-semi group topology on partial homeomorphisms with the weak topology $\tau_{\iota,D}$ on $C_{od}(X,Y)$. It provides a concrete metric $\beta$ that makes $(C_{od}(X,Y),\beta)$ Polish under the standard hypotheses that $X$ is locally compact Hausdorff and second-countable and $Y$ is complete, and also yields an explicit metric for $(\Gamma(X),\tau_{hco})$ in the same setting. The analysis shows that the compact-convergence topology $\tau_{cc}$ coincides with the compact-open topology $\tau_{co}$ in these spaces and relates $D(f)$ to the Fell topology on $CL(X)$, enabling a cohesive metrization framework. These results unify notions from Fell topology, compact convergence, and inverse semigroup topology, with implications for spaces of holomorphic functions on open domains and broader topological-inverse-semigroup theory.
Abstract
We will see how to define the metric $β$, which turns the topological space of continuous functions whose domains are open subsets of a locally compact and second countable space $X$ to values in a polish space $Y$, called $(C_{od}(X,Y),τ_{ι,D})$ into a polish space. In particular, we will present a metric for the inverse semigroup of homeomorphisms of a locally compact, Hausdorff, and second-countable space.