Connectedness of $\mathbb R$ Let $\mathbb R$ denote the real number space, and let sets A and B be closed  and nonempty such that $\mathbb {R} \subset A \cup B $, why is it true that due to the connectedness of $ \mathbb{R} $, $A \cap B \neq \emptyset $?

Your assumptiom is $\mathbb{R} \subset A \cap B$. 
But $\mathbb{R} \neq \emptyset$ so $\emptyset \neq \mathbb{R} \subset A \cap B$ imply $A \cap B \neq \emptyset$. Note that we didn't use connectivity of $\mathbb{R}$.
