What is the most general space where compactly supported continuous functions are uniformly continuous? I managed to prove this for metric spaces but I am interested if it also holds in more general uniform spaces. Thanks
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I think this holds for an arbitrary uniform space.
Indeed, let $(x_\alpha)$ and $(y_\alpha)$ be two nets that approach each other according to the uniform structure (that is, $(x_\alpha, y_\alpha)$ converges to the diagonal). We have to prove that $f(x_\alpha) - f(y_\alpha) \to 0$. Since $f$ is bounded, by choosing subnets we may assume that both $f(x_\alpha)$ and $f(y_\alpha)$ converge to $\xi$ and $\eta$, respectively. Assume that $\xi \neq \eta$, and, say, $\xi \neq 0$. Then $x_\alpha$ is eventually inside the compact support of $f$, hence by a choice of subnets we may assume that $x_\alpha \to x$. But since $(y_\alpha)$ approaches $(x_\alpha)$, $x_\alpha \to x$ implies $y_\alpha \to x$ (this is the definition of topology induced by the uniform structure). Hence $\eta = \xi$.