Extension of a metric defined on a closed subset If $X$ is any metrizable space, $A$ is a closed subset of $X$. 
Let $d$ be a compatible metric on $A$ 
then $d$ can be extended to a compatible metric on $X$.
 A: I cite (with a correction) the beginning of my paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”: 
“The problem of extensions of functions from subobjects to objects in 
various categories was considered by many authors. The classic 
Tietze-Urysohn theorem on extensions of functions from a closed subspace of
a topological space and its generalizations belong to the known results.
Hausdorff [3] showed that every metric from a closed subspace of
a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
[2,6]”.
References 
[2] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional
Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257. 
[3] Hausdorff F. Erweiterung einer Homömorpie, - Fund. Math., 16 (1930) 353-360. 
[4] Isbell J.R. On finite-dimensional uniform spaces, - Pacific
J. of Math., 9 (1959) 107-121.
[6] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof,
Bull. Pol. Ac.:Math., 44 (1996) 267-269.
