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

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 Δ-generated spaces and give results and examples which help to determine when a space is Δ-generated. Our most substantial results show how the D-topology on the function space C(M,N) between smooth manifolds compares to other well-known topologies.

diffeological space, $D$-topology, topologies on function spaces, $\Delta$-generated spaces
Mathematical Subject Classification 2010
Primary: 57P99
Secondary: 58D99, 57R99
Received: 9 August 2013
Revised: 10 March 2014
Accepted: 17 March 2014
Published: 9 October 2014
J. Daniel Christensen
Department of Mathematics
University of Western Ontario
London, ON N6A 5B7
Gordon Sinnamon
Department of Mathematics
University of Western Ontario
London, ON N6A 5B7
Enxin Wu
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna