Show that the following sets are not connected:
(a) A := ;
(b) B := .
For this problem, do I just need to show that A B is the empty set?
If so, I would need to prove that for any x,y A cannot equal any x,y B, right?
You are not asked to show that and are disconnected from each other. (It seems like that's the way you interpreted the problem.) You are asked to show that is disconnected. You are also asked to show that is disconnected.
A set X is disconnected if there exist two open sets, U and V such that:
and
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For (a), take and . It's obvious that . It is also clear that (simply because and vice-versa).
For (b), take and . Then proceed to show that:
1.
2.