Suppose that you can write a set $A \subset \Bbb R^n$ as $A \subset F_1 \cup F_2$ where $F_1, F_2$ are closed, and
$A \cap F_1 \cap F_2=\emptyset $
$F_1 \cap A \neq \emptyset$
$F_2 \cap A \neq \emptyset$
Then show that $A$ is disconnected.
The definition I use in my notes is that you would have to show that this is true for some $F_1, F_2$ being open, so that intuitively I need open sets that "enclose" these closed sets but I am not sure how to proceed. What I have guessed so far would be to use the interior of $F_1, F_2$ but I don't think that will work. I am asking for hints on how to proceed.