A note on approximation and homotopy in $C(X,S^n)$, $n=1,3,7$
For any compact space $X$ and a certain class of $\mathbb R $-subalgebras $R\subseteq C(X,\mathbb R )$ we study the subsets $S^n(R)\subseteq C(X,S^n)$ of maps the coordinate functions of which belong to $R$. The space $C(X,S^n)$ is endowed with the compact-open topology. A classical result of Eilenberg on nullhomotopic maps $X\rightarrow S^1$ is extended to maps $X\rightarrow S^n$, $n=3,7$, by using the multiplication on $S^n$. In case $n=1,3,7$, we make a detailed study of the closure of $S^n(R)$, the homotopy classes of maps in $S^n(R)$ and the interrelation between approximation and homotopy. As an application, some previous results on compact real algebraic sets can be supplemented.