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arXiv · 2011.10733

Topological spaces induced by homotopic distance

Tane Vergili·Ayse Borat
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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.

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Tane Vergili, Ayse Borat. 2020-11-21. Topological spaces induced by homotopic distance. https://arxiv.org/abs/2011.10733

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