The D-topology for diffeological spaces

Abstract

Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the $D$-topology. However, the $D$-topology has not yet been studied seriously in the existing literature. In this paper, we develop the basic theory of the $D$-topology for diffeological spaces. We explain that the topological spaces that arise as the $D$-topology of a diffeological space are exactly the $\Delta$-generated spaces and give results and examples which help to determine when a space is $\Delta$-generated. Our most substantial results show how the $D$-topology on the function space $C^{\infty }(M,N)$ between smooth manifolds compares to other well-known topologies.

v1: 13 pages; v2: 19 pages, includes proofs of conjectures from v1; v3: 18 pages, minor improvements to exposition; v4: 18 pages, minor corrections

Cited by in corpus (14)

• The diffeology of Milnor's classifying space
• Model category of diffeological spaces
• Diffeological vector spaces
• The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces
• Smooth constructions of homotopy-coherent actions
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• Whitney Approximation for Smooth CW Complex
• Transfinite product reduction in fundamental groupoids
• Comparing cubical and globular directed paths
• Tangent spaces of diffeological spaces and their variants
• On the Chern character in Higher Twisted K-theory and spherical T-duality
• Constructing arcs from paths using Zorn's Lemma
• Towards Riemannian diffeology
• Diffeological Principal Bundles and Principal Infinity Bundles