Extension of functions and metrics with variable domains

Abstract

Let $(X,d)$ be a complete, bounded, metric space. For a nonempty, closed subset A of X denote by $C^{⁎}(A\times A)$ the set of all continuous, bounded, real-valued functions on $A\times A$. Denote by

$$C^{\mathrm{†}}=\bigcup \{C^{⁎}(A\times A)|A\text{ is a nonempty closed subset of }X\}$$

the set of all partial, continuous and bounded functions. We prove that there exists a linear, regular extension operator from $C^{\mathrm{†}}$ endowed with the topology of convergence in the Hausdorff distance of graphs of partial functions to the space $C^{⁎}(X\times X)$ with the topology of uniform convergence on compact sets. The constructed extension operator preserves constant functions, pseudometrics, metrics and admissible metrics. For a fixed, nonempty, closed subset A of X the restricted extension operator from $C^{⁎}(A\times A)$ to $C^{⁎}(X\times X)$ is continuous with respect to the topologies of pointwise convergence, uniform convergence on compact sets and uniform convergence considered on both $C^{⁎}(A\times A)$ and $C^{⁎}(X\times X)$.