On continuous linear operators extending metrics

Abstract

Let (X, d) be a complete metric space. We prove that there is a continuous, linear extension operator from the space of all partial, continuous, bounded metrics with closed, bounded domains in X endowed with the Hausdorff metric topology to the space of all continuous, bounded, metrics on X with the topology of uniform convergence on compact sets. This is a variant of the result of Tymchatyn and Zarichnyi for continuous metrics defined on closed, variable domains in a compact metric space. We get a similar result for the case of continuous real-valued functions.