#### Vol. 272, No. 1, 2014

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The $D$-topology for diffeological spaces

### J. Daniel Christensen, Gordon Sinnamon and Enxin Wu

Vol. 272 (2014), No. 1, 87–110
##### 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
Revised: 10 March 2014
Accepted: 17 March 2014
Published: 9 October 2014
##### Authors
 J. Daniel Christensen Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada Gordon Sinnamon Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada Enxin Wu Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna Austria