Extension of a metric from closed subspace of a metrizable space to the whole space

Question

How to prove: Let M be a closed subspace of a metrizable space X. Then, any metric on the subspace M can be extended to a metric on X.

Answer 1

Here is an idea which might work:

Let $\rho$ be a metric on X. Define a metric(!) on X by

$$D(x,y): = \rho(x,M) + \rho(y,M) \quad x,y \notin M, x \neq y$$

$$D(x,y):= 0 \quad x,y \notin M, x =y $$

$$D(x,y):= \rho(x,M) \quad x \notin M, y\in M $$

$$D(x,y) := d(x,y) \quad x,y \in M.$$

In the above $$\rho(x,M) := \inf\{\rho(x,y): y\in M\}$$

for any $x \in X$.

It is useful to note that $\rho(x,M) = 0 \Rightarrow x \in M$ since $M$ is closed in $X$.