locally finite family $ \mathcal{A} $

Question

Hello I have had problems with this exercise

Let $X$ be a topological space and $ \mathcal{A} \subset \mathcal{P}(X) $ a locally finite family such that for any $ A, B \in A $ with $ A \neq B $ and $ \overline{A} \cap \overline{B} = \emptyset $. Prove that for all $ x \in X $ exists an open $ U $ such that $ x \in U $ and $ U $ intersects at most one element of $ \mathcal{A} $

I have managed to prove that there is an open $ U $ such that $ x \in U $ but I have no idea how to prove that this open intersects at most one element of $\mathcal{A}$.

Any hint?

Answer 1

Hint: Spell out the definition of "locally finite".