Completely regular Hausdorff space

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.