Problem:

Suppose that A is a subset of $\displaystyle \mathbb{R}^n$ that fails to be connected(i.e. disconnected), and let U, V be open subset of $\displaystyle \mathbb{R}^n$ that separate A.

Suppose that B is a subset of A that is connected. Prove that either $\displaystyle B \subseteq U $ or $\displaystyle B \subseteq V $

Proof:

Let U, V be disjoint open subsets that separate A, then by definition we know three things:

(i)$\displaystyle A \cap U = \emptyset $ and $\displaystyle A \cap V = \emptyset $

(ii) $\displaystyle (A \cap U ) \cap (A \cap V) = \emptyset$

(iii) $\displaystyle (A \cup U ) \cap ( A \cup V) = A $

Let $\displaystyle B \subset A $ that is connected.

This is what I believe it to be, but don't understand why it's true.

$\displaystyle B \cap U \neq \emptyset \implies B \subseteq U $

$\displaystyle B \cap V \neq \emptyset \implies B \subseteq V $

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.