real analysis - Uniformly continuous bijection from $X$ to the Cantor set

Let $X$ be a metric space, and $C$ be the Cantor set (equipped with the standard topology).

Let $f: X\to C$ be a uniformly continuous function. Assume that $f$ is
a bijection. Does it follow that $f$ is a homeomorphism?

I know that if $X$ is compact, then the answer is yes. This follows from the following theorem:

A continuous bijection from a compact set to a Hausdorff space is
homeomorphism.

I am hoping that the condition "uniform continuity" is strong enough to cover the case even when $K$ is not compact.

I appreciate any help :)

Answer

Let $\rho$ be the discrete metric on the Cantor set $C$, and let $d$ be the usual metric; then the identity map from $\langle C,\rho\rangle$ to $\langle C,d\rangle$ is a uniformly continuous bijection but not a homeomorphism.