How to prove that every uniform space is completely regular?

Question

Wikipedia says "every uniform space is completely regular".

How to prove that every uniform space is completely regular?

Answer 1

It’s fairly trivial if you use the pseudometric definition of uniform space. Let $\mathscr{D}$ be the set of pseudometrics generating the uniform structure. If $U$ is an open nbhd of some $x\in X$, then there are a finite $\{d_1,\dots,d_n\}\subseteq\mathscr{D}$ and an $\epsilon>0$ such that

$$\bigcap_{k=1}^nB_{d_k}(x,\epsilon)\subseteq U\;.$$

Define $$f:X\to\Bbb R:y\mapsto\max_{1\le k\le n}d_k(x,y)\;;$$

$f$ is continuous, and $f(x)=0$. If $y\in X\setminus U$, then $d_k(x,y)\ge\epsilon$ for some $k\in\{1,\dots,n\}$, so $f(y)\ge\epsilon$. Now let

$$g:X\to[0,1]:y\mapsto\frac1\epsilon\min\{f(y),\epsilon\}\;;$$

$g$ is continuous, $g(x)=0$, and $g(y)=1$ for $y\in X\setminus U$.

In any case this is a standard result whose proof can be found in any good topology reference (e.g., Engelking’s General Topology and Willard’s General Topology).