2014
DOI: 10.2140/pjm.2014.272.87
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TheD-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 ∆-generated spaces a… Show more

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Cited by 68 publications
(97 citation statements)
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“…This makes them convenient for many applications, ranging from mathematical physics to homological algebra [Wu] and homotopy theory [CW1]. In [CW2], we studied tangent spaces of diffeological spaces, focusing on the internal tangent space T x (X).…”
Section: Introductionmentioning
confidence: 99%
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“…This makes them convenient for many applications, ranging from mathematical physics to homological algebra [Wu] and homotopy theory [CW1]. In [CW2], we studied tangent spaces of diffeological spaces, focusing on the internal tangent space T x (X).…”
Section: Introductionmentioning
confidence: 99%
“…The descriptions of limits, colimits and function spaces are quite simple, and are concisely described in [CSW,Section 2]. The underlying set of a (co)limit is the (co)limit of the underlying sets.…”
Section: Introductionmentioning
confidence: 99%
“…In fact, being ∆-generated is the same as being R-generated, see [CSW13], and so to define d(X) is is sufficient to consider instead all continuous curves p : R → X. The category of ∆-generated spaces has may desirable properties, even beyond the category of compactly generated spaces, see [Dug].…”
Section: A Comparison With Others Spaces Of Embedded Manifoldsmentioning
confidence: 99%
“…In section Section 2.3 we provide a diffeology for the space ψ d (R m ), that is a collection of smooth plots U → ψ d (R m ) parametrized by finite dimensional smooth manifolds. Topologies induced by diffeologies have been studied before, for example in [CSW13] where they are called the D-topology. In this context we have the following: We also have:…”
Section: A Comparison With Others Spaces Of Embedded Manifoldsmentioning
confidence: 99%
“…For more detailed discussion about the D-topology of diffeological spaces, see [26,Chapter 2] and [8]. Now, let us turn to Frölicher spaces.…”
Section: Both Functors Act As Identity On Arrows)mentioning
confidence: 99%