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Suppose I'm given a manifold $M$ of dimension $n$ and I want to consider the set $$ X = \lbrace H \subset TM | H \textrm{ is a distribution of rank } k \rbrace.$$

What is the usual way to introduce a topology on $X$?

I was thinking, take an open cover by charts and its subordinate partition of unity, then express every distribution as being spanned by $k$ vector fields, then locally compare two distributions, and sum over the partition of unity to get the "distance", but that involves far too many choices, and I'm not sure it would even yield a metric.

EDIT: Maybe some context for this question would be good. I was reading an introductory paper on contact geometry, which said that "the set of all contact structures forms an open subset of the set of all hyperplane fields with the $C^1$-fine topology." So I assume that there also exist $C^2$, $C^3$, ..., $C^{\infty}$ topologies on $X$.

Matija Sreckovic
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  • Distributions of rank $k$ are sections of a bundle (the Grassmannian bundle). In particular, they can be seen as smooth maps, so the space of rank $k$-distributions can be endowed with any of the function space topologies $C^{1},\ldots,C^{k},\ldots,C^{\infty}$. – studiosus Mar 02 '20 at 08:03
  • Can you elaborate on the definition of the function space topologies, or give a reference? Also, if I have a fiber bundle where the fiber and base space are smooth manifolds, then I assume that the total space is also a smooth manifold. Do you know of a place where this claim is proven? I'm asking because I would like to show that the Grassmanian bundle is a smooth manifold. – Matija Sreckovic Mar 04 '20 at 18:25

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