About the space of continuous functions with open domain

TL;DR

This paper introduces a metric that makes the space of continuous functions with open domains into a Polish space, and also provides a metric for the inverse semigroup of homeomorphisms of certain topological spaces.

Contribution

It defines a new metric $eta$ for the space of continuous functions with open domains and establishes its Polish space properties, including for the inverse semigroup of homeomorphisms.

Findings

The space $(C_{od}(X,Y), au_{ ext{iota},D})$ is shown to be Polish.

A specific metric for the inverse semigroup of homeomorphisms is provided.

The results apply to locally compact, second-countable spaces.

Abstract

We will see how to define the metric $\beta$, 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 $(C_{od}(X,Y),\tau_{\iota,D})$ 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.