Suspension pushes up connectivity by one

Statement

Define, for a topological space $X$, the connectivity of $X$ as follows:

• If $X$ is not path-connected, it is $-1$.
• If $X$ is path-connected but not simply connected (i.e., the fundamental group is nontrivial), it is $0$.
• Otherwise, it is the largest $n$ such that the homotopy group ${\pi }_k(X)$ is a trivial group for $1\le k\le n$. If no such largest $n$ exists, set it as $+\mathrm{\infty }$.

The connectivity of the Suspension (?) $SX$ is exactly one more than the connectivity of $X$.

In particular, $X$ is a weakly contractible space if and only if $SX$ is.

Facts used

1. Suspension of any space is path-connected
2. Suspension of path-connected space is simply connected
3. Homology for suspension
4. Hurewicz theorem

Proof

The proof essentially follows from facts (1)-(4).