Problem:
Suppose that A is a subset of that fails to be connected(i.e. disconnected), and let U, V be open subset of that separate A.
Suppose that B is a subset of A that is connected. Prove that either or
Proof:
Let U, V be disjoint open subsets that separate A, then by definition we know three things:
(i) and
(ii)
(iii)
Let that is connected.
This is what I believe it to be, but don't understand why it's true.
If B is a subset of A, then shouldn't U and V disconnect B as well? Thus, B can not be contained in U and B can not be contained in V.
Thank you so much for reading. Any help is greatly appreciated.