Remark

Let $\varphi$ be the canonical mapping of $\mathbf{R}$ onto $\mathbf{T}=\mathbf{R}/\mathbf{Z}$ (TG, V, p. 2). The mapping $f\mapsto f\circ \varphi |\mathbf{I}$ is a homeomorphism of $\mathscr{C}_c(\mathbf{T}; X)$ onto Ω(X), whence, by passing to arcwise connected components, a bijection of the set $[\mathbf{T}; X]$ (III, p. 230) onto the set of free homotopy classes of loops in X.