Let $f:S^n\to C$ and let $K$ be its image. If $n>1$ then the Theorem implies $f\simeq *$.

EDIT:  Clearly this is over-simple and wrong, as Omar Antolín-Camarena has pointed out.  In fact, it looks like $C^N(T)$ would be a compact CW complex, and so there will be surjective continuous functions $S^n \to C^N(T)$ for any $n\geq 1$; then $K = C^N(T)$.