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!

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    $\begingroup$ Note on terminology: $A\times B$ is the Cartesian product. The cross product is something entirely different. $\endgroup$ – Tim Raczkowski Nov 5 '15 at 1:36
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    $\begingroup$ In $\Bbb R$ the intersection of two connected sets is connected (possibly empty, of course), but there are simple counterexamples in $\Bbb R^2$. $\endgroup$ – David C. Ullrich Nov 5 '15 at 1:41
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    $\begingroup$ For (iii) you may notice that if $(x,b) \in A \times B$, then for every $x \in A$ the subspace $\{x \} \times B \cup A \times \{ b \}$ is connected since $ (A \cup \{b\} ) \cap (\{x\} \cup B ) = (x,b)$... $\endgroup$ – gangrene Nov 5 '15 at 1:53
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    $\begingroup$ Here's Wikipedia's picture, if you'll excuse the French. $\endgroup$ – Akiva Weinberger Nov 5 '15 at 3:08

i) Not true, as you said, though union of two sets with one point in common is connected.

ii) Not true. Take two cheetos reflected.

iii) True. You can use the characterization that $X$ is connected iff every continuous function from $X$ to the discrete space $\{0,1\}$ is constant. Or you can simply find a smart way of showing that $X \times Y$ is the union of pairwise non-disjoint connected spaces.

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    $\begingroup$ Thank you very much, I appreciate your help! I think the phrase "two cheetos reflected" is going to be a favorite of mine for a long time. $\endgroup$ – Iff Nov 5 '15 at 3:54

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