TITLE: Intersection of Simply-Connected Sets
QUESTION [5 upvotes]: Let $U,V$ be two simply connected subsets of a topological space.
Prove or disprove:
$U \cap V$ is simply connected.

REPLY [8 votes]: Let $S^1$ be the circle in $\mathbb R^2$, $U=\{(x,y)\in S^1: x\geq 0\}$ and $V=\{(x,y)\in S^1: x\leq 0\}$. Then $U$ is the right half of a circle and $V$ is the left half, both of which are simply connected. What is $U\cap V$?