If every continuous function $f:X\to\mathbb{R}$ (where $X$ is a subset of a compact metric space), is uniformly continuous, then am I right to assume that $X$ is compact as well?
I think it should but I'm not sure if I am correct. Like if $X$ is a subset of a compact metric space then shouldn't $X$ itself the domain be compact as well if $f$ is uniformly continuous?