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.

Simultaneous extensions of metrics and ultrametrics of high power

“…For a metric space (X, d), we denote by exp(X) the space of all nonempty compact subsets of X equipped with the Hausdorff distance. By Michael's zero-dimensional selection theorem [37,Theorem 2], for a complete ultrametric space (X, d), Tymchatyn-Zarichnyi [50] constructed a map R : X × exp(X) → X such that R(x, A) ∈ A for all x ∈ X, and x ∈ A implies R(x, A) = x. Stasyuk-Tymchatyn [48] proved the existence of uniformly continuous R : X × exp(X) → X satisfying the conditions mentioned above.…”

“…For a compact ultrametrizable X, Tymchatyn-Zarichnyi [50] constructed maps from the set of all continuous ultrametrics defined on closed subsets of X into the set of all continuous ultrametrics defined on X, which is continuous with respect to the Vietoris topology. Stasyuk-Tymchatyn [48] constructed such a map for a complete ultrametrizable space. The author does not know whether we can constructed a Tymchatyn-Zarichnyi type map from A∈exp(X) UMet(X; R ≥0 ) into UMet(X; R ≥0 ) or not.…”

Extending proper metrics

“…For variants of Hausdorff's extension theorem, see, for example, [19], [12], [11]. For extensions of ultrametrics (non-Archimedean metrics), see [42], [39], [20], [21], [23], and [22].…”