Dimension in diffeology
Jun 1, 2006, update September 16, 2007
 
This paper generalizes to diffeology the usual notion of dimension defined for manifolds. But diffeological spaces may own singularities, it’s why the “dimension map”, introduced also here, is more appropriate to diffeology.The dimension is a diffeological invariant and the two following examples show how this notion of dimension works.
Example 1. The quotient ∆n = Rn/O(n,R) is homeomorphic to the interval [0,∞[ but has dimension n (Precisely, the dimension of ∆n is n in 0, and 1 elsewhere). Example 2. The half-line ∆∞ = [0,∞[, equipped with the subset diffeology of R, has dimension ∞ (Precisely, the dimension of ∆∞ is ∞ in 0, and 1 elsewhere). By the way, since the dimension function is a diffeological invariant, this shows without any topological argument that diffeomorphisms of these spaces fix necessarily the origin 0.  Indagationes Mathematicae, 18(4) 2007.http://www.elsevier.com/wps/find/journaldescription.cws_home/505620/description#descriptionshapeimage_3_link_0
First draft (extended version) june 2006