2
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ Try drawing a picture of sets like this in $\mathbb{R}^2$ and going from there. I think that's most likely the best way towards intuitively understanding how the solution works for this problem. $\endgroup$ – Alex Nolte May 27 '18 at 0:09
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.