Topological groups are completely regular

Have a look at this:

http://www.math.wm.edu/~vinroot/PadicGroups/519probset1.pdf"

Problem 2 is what you want.

Every uniform space $(X,\mathcal D)$ is completely regular.

sketch of a proof: Suppose $F$ is closed in $(X,\mathcal D)$ and $p\in F^c$. There's a pseudometric uniformity $P$ on $X$, such that: $$\mathcal D=\mathcal D_P=\bigcup_{d\in P}\mathcal D_d$$ Where $\mathcal D_d$ is the usual uniformity by the pseudometric $d:X^2\to [0,\infty)$.

For each $d\in P$, define $$f_d:X\to\Bbb R$$ $$f_d(x)=\inf_{c\in F}d(c,x)$$ and $$g_d:X\to \Bbb R$$ $$g_d(x)=d(p,x)$$ $f_d$ and $g_d$ are continuous. It's not hard to prove there's some $d_0\in P$ with $$(\forall a\in X)(f_{d_0}(a)\ne 0\text{ or } g_{d_0}(a)\ne 0)$$ This may help. Define $$h:X\to [0,1]$$ $$h(x)=\frac{g_{d_0}(x)}{g_{d_0}(x)+f_{d_0}(x)}$$ $h$ is continuous and $$h(p)=0,\quad h(F)=\{1\}$$

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Edit:

linked