Topological spaces induced by homotopic distance

TL;DR

This paper explores the topology induced by homotopic distance as a pseudometric on mapping spaces, specifically analyzing the space of maps from the circle to itself, and establishes properties like fixed homotopic distances and non-compactness.

Contribution

It studies the topology induced by homotopic distance on mapping spaces and characterizes the space of circle maps, including its non-compactness.

Findings

Homotopic distance defines a pseudometric on mapping spaces.

In $ ext{Map}(S^1,S^1)$, all maps have homotopic distance 1.

The space $ ext{Map}(S^1,S^1)$ is not compact.

Abstract

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.