I have been studying the article "Hyperspaces of finite subsets which are homeomorphic to $\aleph_0$-dimensional linear metric spaces" by Curtis and Nguyen. There they say that the hyperspace of finite subsets of $\mathbb{S}^1$ with no more than three points is simply connected and that it is a result proved by Bott. They also show a homotopy that will be enough to prove the lemma instead of proving the simple connectedness. However, I couldn't understand the homotopy construction, nor find the result mentioned. Here is the lemma
I would like some help on this homotopy or to know where can I find this result by Bott. I even tried variations of this one, changing the domain for $t$, and changing that intersection (which doesn't make sense to me), but I couldn't go anywhere.