I am given that $(X,d)$ is a metric space, and that $A,B \subseteq X$ where $A,B$ are connected.
I have to show which of the following are necessarily true:
i) $A \cup B$ is connected
ii) $A \cap B$ is connected
iii) $A \times B$ is connected
Definition of connected: Let $(X,d)$ be a metric space. Then $(X,d)$ is connected provided $X$ is not the union of two nonempty disjoint open sets. A subset $S$ of $X$ is connected provided $(S,d)$ is a connected space.
Informally, I believe the union is not connected, since a counterexample could be two intervals in $\Bbb R$, $A = (1,2)$ and $B = (3,4)$. I think that perhaps the intersection would be true, but I have no clue as to what the cross product would be. I'm having a hard time picturing the shape for iii).
Any help about how to proceed would be appreciated!