Description of a set in the Baire space.

Question

I have the following exercise.

In the Baire space, $\omega ^ \omega$ ,describe a set $A$ such that $int(A)\neq \emptyset$ and $A\neq \bar{A}$

I recently just got to know the Baire space, it is difficult for me to understand it a bit and I have come across this exercise. Could you please help me with an example. Thank you.

Edit:

I have considered the following set: $A=\{ x\in \omega ^ \omega: x\nleq f\}$ And I have shown that it is open. But I have the doubt, that for example in the discrete metric all open is closed at the same time, and I cannot see how to show that this set is not closed

Answer 1

HINT: Consider the point $f$ itself: clearly $f\notin A$, but you should be able to show that $f\in\operatorname{cl}A$. It may help to prove that the sets $$B_n=\{x\in\omega^\omega:x(k)=f(k)\text{ for all }k\le n\}$$ for $n\in\omega$ are a local base at $f$, if you don’t already know this fact.