Disconnected sets in $\Bbb R^n$ and closed sets.

Question

Problem:

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.

Answer 1

Hint: show that the same list of properties is true for $F_1^c$ and $F_2^c$.