A metric for the space of submanifolds of Galatius and Randal-Williams

Abstract

Galatius and Randal-Williams defined a topology on the set of closed submanifolds of ${\mathbb R}^n$. B\"okstedt and Madsen proved that a $C^1$-version of this topology is metrizable by showing that it is regular and second countable. Using that the scanning map of a topological sheaf on manifolds is an embedding, we give an explicit metric to the space considered by B\"okstedt and Madsen. Then, we compare this topology with the Fell topology and we use the Hausdorff distance to give another metric to the space of Galatius and Randal-Williams.