Is it true that two subsets of a metric space are disconnected if and only if there does not exist a Cauchy sequence with all points in one subset and its limit in the other?
It seems right for simple cases, though I can't come up with a rigorous proof.I'm having a hard time with the concept of connectedness.
Edit:
Is it true that a metric space is connected if and only if for each proper subset $A$, there exists another subset $B$, connected to $A$, such that $A \cap B = \emptyset$?
Thanks.