Suspension of path-connected space is simply connected

Statement
Suppose $$X$$ is a fact about::path-connected space. Denote by $$SX$$ the fact about::suspension of $$X$$. Then, $$SX$$ is a fact about::simply connected space.

Applications

 * n-sphere is simply connected for n greater than 1: The $$n$$-sphere $$S^n$$ is the suspension of the $$(n - 1)$$-sphere $$S^{n-1}$$.

Facts used

 * 1) uses::Union of two simply connected open subsets with path-connected intersection is simply connected, which in turn uses the uses::Seifert-van Kampen theorem