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 $$+\infty$$ (when this occurs, we say that $$X$$ is a weakly contractible space).

The connectivity of the fact about::suspension $$SX$$ is exactly one more than the connectivity of $$X$$.

In particular, $$X$$ is a uses property satisfaction of::weakly contractible space  if and only if $$SX$$ is.

Facts used

 * 1) uses::Suspension of any space is path-connected
 * 2) uses::Suspension of path-connected space is simply connected
 * 3) uses::Homology for suspension
 * 4) uses::Hurewicz theorem

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