This definition is silent on which category the real line is being considered as an object of. For instance, if is regarded as a topological space (hence an object in the category Top), then the topology on is Euclidean topology the real line with itself where is some natural number. Another possibility is to regard as a smooth manifold (hence an object in the category Diff). The Cartesian space with its standard topology (and sometimes smooth structure) is also called real -dimensional space (distinguish from “real -dimensional vector space” which is only isomorphic to it as a vector space).
Cartesian spaces carry plenty of further canonical structure:
Sometimes one is interested in allowing to take other values, in which case one wants a product in some category that might not be the Cartesian product on underlying sets.
For example, if one is studying Cartesian spaces as inner product spaces, then one might want an -dimensional Cartesian space to be the -dimensional Hilbert space , which is a proper subset of the cartesian product .
In fact, in there is no choice:
This was shown in (Stallings).
In the analog of this statement is false. One says that on there exist exotic smooth structures.
In dimension for we have:
See the first page of (Ozols) for a list of references.
In dimension 4 the analog statement fails due to the existence of exotic smooth structures on .
There are various slight variations of the category that one can consider without changing its basic properties as a category of test spaces for generalized smooth spaces. A different choice that enjoys some popularity in the literature is the category of open (contractible) subsets of Euclidean spaces. For more references on this see diffeological space.
in detal in section 5 of
and briefly mentioned in example 2) on p. 191 of
following the original article
With an eye towards Frölicher spaces the site is also considered in section 5 of