Let $\sum X$ be the reduce suspension of $X$.
Freudenthal Suspension Theorem: X is nondegenerately based and (n−1)-connected, where n ≥ 1. Then $\sum: \pi_q(X) \to \pi_q(\sum X)$ is a bijection if $q < 2n−1$ and a surjection if $q = 2n − 1$.
Note: nondegenerately based means the inclusion of the based point $\ast \hookrightarrow X $ is a cofibration.
We know that based on Freudenthal Suspension Theorem, for nondegenerately based path connected space X, $\sum X$ is simply connected.
My question:
Is there any example such that: X is path connected but not nondegenerately based and $\sum X$ is not simply connected.
As pointed out in comments at math.stackexchange.com/questions/195076/…, Tom Goodwillie described an example: let $X$ be the unreduced cone $CZ$ on the set $Z=\{ 0,1,1/2,1/3,1/4,...\}$. See lehigh.edu/~dmd1/tg26, lehigh.edu/~dmd1/tg27, and lehigh.edu/~dmd1/tg28.