Smooth distributions on subcartesian spaces are globally finitely generated

TL;DR

This paper proves that connected subcartesian spaces can be embedded in Euclidean space and that smooth generalized distributions and subbundles on these spaces are globally finitely generated, extending classical results.

Contribution

It establishes the embedding of connected subcartesian spaces into Euclidean space and demonstrates finite generation of distributions and subbundles on these spaces.

Findings

Connected subcartesian spaces admit Euclidean embeddings.

Smooth generalized distributions are globally finitely generated.

Smooth subbundles are globally finitely generated.

Abstract

We prove that a connected subcartesian space admits embedding in a Euclidean space. The Whitney Embedding Theorem is then stated as a corollary of our result. Based on the above result together with the theory of distribution on smooth manifolds, we show that smooth generalized distributions on connected subcartesian spaces are globally finitely generated. We also show that smooth generalized subbundles of vector bundles on connected subcartesian spaces are globally finitely generated.