Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topological spaces induced by homotopic distance   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.

36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th