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.