Suppose are connected subsets of some metric space . Prove that if the is connected.
So suppose is not connected.
Then where and are disjoint and clopen. So and . Also and .
So . Now what? How do you conclude that ?
I have no idea how your textbook/instructor requires you to prove connectivity.
The usual definition simply says that X is connected if and only if there do no exist two sets C & D such that .
Put simply: X is not a subset of the union of two sets, each having a nonempty intersection with X, and neither contains a point or a limit point of the other. That is known as a separation of X.
It seems that you are to use some other approach to connectivity.