Suspension of path-connected space is simply connected

Statement

Suppose $X$ is a Path-connected space (?). Denote by $SX$ the Suspension (?) of $X$. Then, $SX$ is a Simply connected space (?).

Related facts

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

Proof

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