Topological spaces induced by homotopic distance

Homotopic distance $\text{D}$ as introduced in \cite{MVML} can be realized as a pseudometric on $\mathrm{M}\mathrm{a}\mathrm{p}(X,Y)$. In this paper, we study the topology induced by the pseudometric $\text{D}$. In particular, we consider the space $\mathrm{M}\mathrm{a}\mathrm{p}(S^1,S^1)$ 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 $\mathrm{M}\mathrm{a}\mathrm{p}(X,Y)$ is still an open problem, it can be shown that $\mathrm{M}\mathrm{a}\mathrm{p}(S^1,S^1)$ is not compact.