Exercise 3, § 11

Let $A, B$ be two subsets of a topological space $X$.

a) Show that if $A$ and $B$ are closed in $X$, and $A \cup B$ and $A \cap B$ are connected, then $A$ and $B$ are connected. Show by an example that this statement is no longer necessarily true when one of the sets $A, B$ is not closed.

b) Suppose that $A$ and $B$ are connected and $\overline{A} \cap B \neq \varnothing$. Show that $A \cup B$ is connected.