The D-topology for diffeological spaces

TL;DR

This paper develops the theory of the $D$-topology for diffeological spaces, showing its relation to $ riangle$-generated spaces and comparing it to other topologies on function spaces.

Contribution

It establishes the basic properties of the $D$-topology, characterizes $ riangle$-generated spaces as $D$-topologies, and compares the $D$-topology on function spaces to existing topologies.

Findings

The $D$-topology characterizes $ riangle$-generated spaces.

The $D$-topology on $C^{inity}(M,N)$ is compared to other topologies.

Examples illustrate when a space is $ riangle$-generated.

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.