If $K$ is an $n$-skeleton of an $(n+1)$-simplex, then $|K| \cong S^n$

Question

If $K$ is an $n$-skeleton of an $(n+1)$-simplex, then $|K| \cong S^n$, where $\cong$ means homeomorphism.

One can trivially come up with a homeomorphism in low dimensions, but anything past $3$ dimensions I have no idea how to find the homeomorphism.

I tried to show that $\dot \Delta^{n+1} \cong S^n$ but ran into trouble, as if $h : |K| \rightarrow S^n$ then I tried to show the homeomorphism via mapping vertexes $\{p_i \in \Delta^{n+1}\} \rightarrow \{a_i \in S^n\}$ and then mapping all $[p_i, \dots ]$ to the surfaces of $[a_i, \dots] \subseteq S^n$ but I couldn't show that this was a homeomorphism.

Anyone have any ideas?

Answer 1

Project from the barycentre of the simplex to the circumscribing sphere.