Topological spaces induced by homotopic distance

Topological complexity which plays an important role in motion planning problem can be generalized to homotopic distance D as introduced in [6]. In this paper, we study the homotopic distance and mention that it can be realized as a pseudometric on Map(X, Y ). Moreover we study the topology induced by the pseudometric D. In particular, we consider the space Map(S^1, S^1) and use the non-compactness of it to talk about the non-compactness of Map(X, Y ).