0
$\begingroup$

It is known that $Y \cup A$ and $Y \cup B$ are connected given that $X$ is a connected space, $Y$ is a connected subspace of $X$, and that $X-Y$ is separated by $A$ and $B$. Is it true that $A$ and $B$ are connected as well?

$\endgroup$
2
$\begingroup$

No, this is false. Let $X \subset \Bbb{R}^2 $ be the union of two closed discs $D_1, D_2$ whose boundaries intersect in a single point $p$. Let $L$ be the line connecting $p$ to the centers of both discs, and $Y = X \cap L$. If you let $A = D_1 /Y$, $B = D_2/Y$ then $Y$, $Y \cup A$ and $Y \cup B$ are all connected and $X/Y$ is separated by $A$ and $B$, but both $A$ and $B$ are not connected.

$\endgroup$

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.