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 Simply connected space (?)s and in particular Path-connected space (?)s.
• The intersection $W=U\cap V$ is a non-empty path-connected space.

Then, $X$ is a Simply connected space (?) (and in particular, a path-connected space).

Facts used

1. 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.