  Homotopic distance $\D$ as introduced in \cite{MVML} can be realized as a
pseudometric on $\mathrm{Map}(X,Y)$. In this paper, we study the topology
induced by the pseudometric $\D$. In particular, we consider the space
$\mathrm{Map}(S^1,S^1)$ and show that homotopic distance between any two maps
in this space is 1. Moreover, while a general proof of the non-compactness of
the space $\mathrm{Map}(X,Y)$ is still an open problem, it can be shown that
$\mathrm{Map}(S^1,S^1)$ is not compact.
