Prove that if F closed and of first category in a complete metric space, then F is nowhere dense.

Question

Prove that if F closed and of first category in a complete metric space, then F is nowhere dense.

I have tried it but i can't find any clues.

Answer 1

$F=\cup_n F_n$ where $F_n$ is nowehere dense since $F$ is a first category set. Let $\bar L$ be the adherence of $L$, since $F_n\subset F, \bar F_n\subset\bar F=F$, we deduce that $\bar F=F=\cup_n\bar F_n$. The theorem of Baire implies that $F$ is nowhere dense.