2022/12/13 by Yael Karshon, Karshon, Yael, Jordan Watts +1
Decision Sciences · Mathematics · #Advanced Topology and Set Theory #Differential Geometry (math.DG) #FOS: Mathematics #Fuzzy and Soft Set Theory #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2212.06917
openalex publication_date 2022/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are several notions of a smooth map from a convex set to a cartesian space. Some of these notions coincide, but not all of them do. We construct a real-valued function on a convex subset of the plane that does not extend to a smooth function on any open neighbourhood of the convex set, but that for each k extends to a Ck function on an open neighbourhood of the convex set. It follows that the diffeological and Sikorski notions of smoothness on convex sets do not coincide. We show that, for a convex set that is locally closed, these notions do coincide. With the diffeological notion of smoothness for convex sets, we then show that the category of diffeological spaces is isomorphic to the category of so-called exhaustive Chen spaces.