Union of two simply connected open subsets with path-connected intersection is simply connected

Statement
Suppose $$X$$ is a topological space with two non-empty open subsets $$U$$ and $$V$$ such that:


 * $$X = U \cup V$$.
 * Both $$U$$ and $$V$$ are fact about::simply connected spaces and in particular fact about::path-connected spaces.
 * The intersection $$W = U \cap V$$ is a non-empty path-connected space.

Then, $$X$$ is a fact about::simply connected space (and in particular, a path-connected space).

Facts used

 * 1) uses::Seifert-van Kampen theorem

Applications

 * n-sphere is simply connected for n greater than 1

Proof
The statement follows directly from the Seifert-van Kampen theorem. Both $$\pi_1(U)$$ and $$\pi_1(V)$$ are trivial, so we get $$\pi_1(X)$$ is an amalgamated free product of two trivial groups, hence it must be trivial.