Is it true that, $A,B\subset X$ are completely seprated iff their closures are? If $A,B\subset X$ and $\overline{A}, \overline{B}$ are completely seprated, so also are $A,B$. since  $A\subset \overline{A}$, $B\subset \overline{B}$ then, $f(A)\subset f(\overline{A})=0$ and $f(B)\subset f(\overline{B})=1$ for some continuous function $f:X\to [0,1]$. but the converse is true?

HINT: Suppose that $A$ and $B$ are completely separated, and let $f:X\to[0,1]$ be a continuous function such that $f(x)=0$ for all $x\in A$ and $f(x)=1$ for all $x\in B$. Since $f$ is continuous, $f^{-1}[\{0\}]$ is closed, and certainly $A\subseteq f^{-1}[\{0\}]$, so ... ?
