nLab
cartesian space

A Cartesian space is a space of the form ${ℝ}^n$, the $n$-fold Cartesian product of the real line $ℝ$, where $n$ is some natural number (possibly zero).

In particular:

• ${ℝ}^0$ is the point,
• ${ℝ}^1$ is the real line itself,
• ${ℝ}^2$ is the real plane, which may be identified (in two canonical ways) with the complex plane $ℂ$.

Note that a Cartesian space has maximal structure; it comes equipped with a system of coordinates. Usually we are interested only in some of this structure; what is useful is that the category CartSp of cartesian spaces and (appropriately) structure-preserving maps is a small category equivalent to the (usually large) category of all finite-dimensional spaces with the structure in question. This holds at least in the following examples:

• vector spaces,
• affine spaces,
• normed vector spaces,
• inner product spaces,
• Euclidean spaces.

$\mathrm{Cart}\mathrm{Sp}$ can also be understood, for appropriate morphisms, as a category of models for at least the following examples:

• topological manifolds,
• smooth spaces,
• Riemannian manifolds.

Sometimes one is interested in allowing $n$ 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 ${\aleph }_0$-dimensional Cartesian space to be the ${\aleph }_0$-dimensional Hilbert space $l^2$, which is a proper subset of the cartesian product ${ℝ}^{{\aleph }_0}$.