= Completely regular Hausdorff space
= Tychonoff space
{c}
{synonym}
A <Hausdorff space> is completely regular if a point $x$ outside a <closed set> $F$ can be separated from $F$ by a <continuous function> $u:X\to[0,1]$ with $u(x)=1$ and $u|_F=0$. These functions determine the original <topology>: for any neighborhood $O$ of $x$, choose $F=X\setminus O$ to obtain $x\in\{u>1/2\}\subseteq O$. This is exactly the point-separation property needed for the evaluation embedding into the weak-star dual of bounded continuous functions.
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