The $D$-topology for diffeological spaces

J. Daniel Christensen, Gordon Sinnamon and Enxin Wu

Vol. 272 (2014), No. 1, 87–110

DOI: 10.2140/pjm.2014.272.87

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 $C^{\infty }\left(M,N\right)$ between smooth manifolds compares to other well-known topologies.

Keywords

diffeological space, $D$-topology, topologies on function spaces, $\Delta$-generated spaces

Mathematical Subject Classification 2010

Primary: 57P99

Secondary: 58D99, 57R99

Milestones

Received: 9 August 2013

Revised: 10 March 2014

Accepted: 17 March 2014

Published: 9 October 2014