# Proving the union and intersection of connected subsets is also connected.

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!

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

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.