On classification of spaces of continuous $S^1$-valued functions on polihydrons

In this paper, the spaces of continuous $S^1$-valued functions $C_p(X,S^1)$ are considered. It is proved that if $X$ is a $n$-dimensional polihydron and $S^1$ is a circle which is considered as a topological group, then the topological group $C_p(X,S^1)$ is topologically isomorphic to $C_p(\Delta_n,S^1)$, where $\Delta_n$ is an $n$-dimensional simplex, $n\geqslant1$.