A constructive proof for the simple connectedness of finite subset spaces

The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces ${\mathrm{R}\mathrm{a}\mathrm{n}}_{⩽n}(X)$ of the Ran space given by all subsets of $X$ of size at most $n$, and their first homotopy groups. These groups are known to be trivial for $n⩾3$ when $X$ is a path connected CW-complex, though the proofs are not constructive. We show that the induced map ${\pi }_1({\mathrm{R}\mathrm{a}\mathrm{n}}_{⩽n}(X))\to {\pi }_1({\mathrm{R}\mathrm{a}\mathrm{n}}_{⩽n+2}(X))$ is trivial for all positive integers $n$, by explicitly drawing the path homotopies that contract any loop in $X$ to a point. From this we get a constructive proof for the triviality of ${\pi }_1({\mathrm{R}\mathrm{a}\mathrm{n}}_{⩽n}(X))$, for all $n⩾4$.