Extension of functions and metrics with variable domains

Let (X,d) be a complete, bounded, metric space. For a nonempty, closed subset A of X denote by C⁎(A×A) the set of all continuous, bounded, real-valued functions on A×A. Denote byC†=⋃{C⁎(A×A)|A 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† endowed with the topology of convergence in the Hausdorff distance of graphs of partial functions to the space C⁎(X×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×A) to C⁎(X×X) is continuous with respect to the topologies of pointwise convergence, uniform convergence on compact sets and uniform convergence considered on both C⁎(A×A) and C⁎(X×X).