$C$-embedding in uniform spaces

Question

Every Hausdorff uniform space $X$ has a Hausdorff completion $C_X$. Is it true that $X$ is $C$-embedded in $C_X$? How about the completion with respect to its finest uniformity $\mu_X$?

Answer 1

Let $X=\Bbb R\setminus\{0\}$. $\Bbb R$ is a Hausdorff completion for $X$. The function $$f:X\to \Bbb R$$ $$f(x)={1\over x}$$ is continuous. But it cannot be extended to a continuous function on $\Bbb R$. So the uniform space $X$ cannot be $C$-embedded in any of its Hausdorff completions.

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Let $(X,\mathcal D)$ be a fine uniform space with a Hausdorff completion $(Y,\mathcal E)$. Then any continuous function $$f:X\to \Bbb R$$ is uniformly continuous. So, because $\Bbb R$ is complete, it can be extended to a (uniformly) continuous function: $$g:Y\to \Bbb R$$ So $X$ can be $C$-embedded in $Y$.