If X is a set, a diffeology on X is a set of maps (called plots) from open subsets of some Euclidean space to X such that the following hold:
A set together with a diffeology is called a diffeological space.
A map between diffeological spaces is called differentiable if and only if composing it with every plot of the first space is a plot of the second space. It is a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable.
The diffeological spaces, together with differentiable maps as morphisms, form a category. The isomorphisms in this category are just the diffeomorphisms defined above.
A diffeological space has the Dtopology: the finest topology such that all plots are continuous.
If Y is a subset of the diffeological space X, then Y is itself a diffeological space in a natural way: the plots of Y are those plots of X whose images are subsets of Y.
Every smooth (i.e. C^{∞}) manifold has a diffeology: the one where the plots are the smooth maps from open subsets of Euclidean spaces to the manifold. In particular, every open subset of R^{n} has a diffeology.
The smooth manifolds with smooth maps can then be seen as a full subcategory of the category of diffeological spaces.
A diffeological space where every point has a Dtopology neighbourhood diffeomorphic to an open subset of R^{n} (where n is fixed) is the same as the diffeology generated as above from a manifold structure.
The notion of a generating family, due to Patrick Iglesias, is convenient in defining diffeologies: a set of plots is a generating family for a diffeology if the diffeology is the smallest diffeology containing all the given plots. In that case, we also say that the diffeology is generated by the given plots.
If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X/~ has the diffeology generated by all compositions of plots of X with the projection from X to X/~. This is called the quotient diffeology. Note that the quotient Dtopology[?] is the Dtopology of the quotient diffeology.
This is a way to easily get nonmanifold diffeologies. For example, the real numbers R are a diffeological space (they are a manifold). R/(Z + αZ), for some irrational α, is the irrational torus. It has a diffeology, but the Dtopology for it is the trivial topology.
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