nLab cartesian space

Idea

A Cartesian space is a finite Cartesian product of the real line $\mathbb{R}$ with itself. Hence, a Cartesian space has the form $\mathbb{R}^n$ where $n$ is some natural number (possibly zero).

What exactly this means depends on which category the real line $\mathbb{R}$ is being considered as an object of.

For instance, if $\mathbb{R}$ is regarded as a topological space (hence an object in the category Top), then $\mathbb{R}^n$ is a Euclidean space. Another possibility is to regard $\mathbb{R}$ as a topological manifold, differentiable manifold or smooth manifold (hence an object in the categories TopMfd or Diff), in which case $\mathbb{R}^n$ is the archetypical such manifold of dimension $n$ (the basic coordinate chart).

The Cartesian space $\mathbb{R}^n$ with its standard topology/smooth structure is also called real $n$-dimensional space.

One could also regard $\mathbb{R}$ as the real vector space of dimension 1, in which case $\mathbb{R}^n$ would be the real vector space of dimension $n$ (maybe understood as equipped with the linear basis canonically induced by this presentation), hence the “real $n$-dimensional vector space”. However, a vector space would typically not be referred to as a “Cartesian space”.

But in between the last two examples, regarding $\mathbb{R}^n$ as an affine space makes it the basis of “analytic geometry” in the sense originally due to René Descartes (Cartesian geometry?), and this is where the term “Cartesian space” originates from.

One might also speak of the complexified cartesian space $\mathbb{C}^n$ as a complex manifold, or indeed of the cartesian space $K^n$ for any field $K$, maybe regarded as an analytic space, though this is not common terminology.

Example

In particular:

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

Remark

Cartesian spaces carry plenty of further canonical structure:

• It is canonically a metric space and the Euclidean topology is the corresponding metric topology.

• There is a canonical smooth structure on $\mathbb{R}^n$ that makes it a smooth manifold.

• A Cartesian space is canonically avector space over the field of real numbers.

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 $\mathbb{R}^{\aleph_0}$.

Properties

Topological structure

The open n-ball is homeomorphic Cartesian space $\mathbb{R}^n$

$$\mathbb{B}^n \simeq \mathbb{R}^n \,.$$

• topological invariance of dimension

Smooth structures

For all $n \in \mathbb{N}$, the open n-ball with its standard smooth structure is diffeomorphic to the Cartesian space $\mathbb{R}^n$ with its standard smooth structure

$$\mathbb{B}^n \simeq \mathbb{R}^n \,.$$

In fact, in $d \neq 4$ there is no choice:

Theorem

For $n \in \mathbb{N}$ a natural number with $n \neq 4$, there is a unique (up to isomorphism) smooth structure on the Cartesian space $\mathbb{R}^n$.

This was shown in (Stallings).

Theorem

In $d = 4$ the analog of this statement is false. One says that on $\mathbb{R}^4$ there exist exotic smooth structures.

Theorem

In dimension $d \in \mathbb{N}$ for $d \neq 4$ we have:

every open subset of $\mathbb{R}^d$ which is homeomorphic to $\mathbb{B}^d$ is also diffeomorphic to it.

See the first page of (Ozols) for a list of references.

Remark

In dimension 4 the analog statement fails due to the existence of exotic smooth structures on $\mathbb{R}^4$.

The category of Cartesian spaces

See CartSp.

Related concepts

• super Cartesian space

• polydisk

• affine space

In complex geometry for purposes of Cech cohomology the role of Cartesian spaces is played by Stein manifolds.

• topological invariance of dimension

References

Named after René Descartes.

• John Stallings, The piecewise linear structure of Euclidean space , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. (pdf)

• V. Ozols, Largest normal neighbourhoods , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (jstor)

There are various slight variations of the category $CartSp$ 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.

The site $ThCartSp$ of infinitesimally thickened Cartesian spaces is known as the site for the Cahiers topos. It is considered

in detal in section 5 of

• Anders Kock, Convenient vector spaces embed into the Cahiers topos (numdam)

and briefly mentioned in example 2) on p. 191 of

• Anders Kock, Synthetic differential geometry (pdf)

following the original article

• Eduardo Dubuc, Sur les modeles de la geometrie differentielle synthetique (numdam).

With an eye towards Frölicher spaces the site is also considered in section 5 of

• Hirokazu Nishimura, Beyond the Regnant Philosophy of Manifolds (arXiv:0912.0827)