Additional consequence of Urysohn's Lemma?

Question

Urysohn's Lemma states that for any normal topological space $X$ and closed disjoint subsets $A,B\subset X$, we can find a continuous function $f\colon X\to[0,1]$ such that $f|_{A}=0$ and $f|_{B}=1$. But can we always find $f$ such that $f>0$ on $X\setminus A$? Any suggestions would be greatly appreciated!