nLab
cartesian space

Context

Topology

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

Definition

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

This definition is silent 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 the topology on n\mathbb{R}^n is Euclidean topology the real line \mathbb{R} with itself where nn is some natural number. Another possibility is to regard \mathbb{R} as a smooth manifold (hence an object in the category Diff). The Cartesian space n\mathbb{R}^n with its standard topology (and sometimes smooth structure) is also called real nn-dimensional space (distinguish from “real nn-dimensional vector space” which is only isomorphic to it as a vector space).

One may also speak of the complexified cartesian space n\mathbb{C}^n, or indeed of the cartesian space K nK^n for any field KK, or indeed for any set KK, or indeed for any object KK of any cartesian monoidal category.

Example

In particular: * 0\mathbb{R}^0 is the point, * 1\mathbb{R}^1 is the real line, * 2\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:

Sometimes one is interested in allowing nn 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 0\aleph_0-dimensional Cartesian space to be the 0\aleph_0-dimensional Hilbert space l 2l^2, which is a proper subset of the cartesian product 0\mathbb{R}^{\aleph_0}.

Properties

Topological structure

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

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

Smooth structures

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

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

In fact, in d4d \neq 4 there is no choice:

Theorem

For nn \in \mathbb{N} a natural number with n4n \neq 4, there is a unique (up to isomorphism) smooth structure on the Cartesian space n\mathbb{R}^n.

This was shown in (Stallings).

Theorem

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

Theorem

In dimension dd \in \mathbb{N} for d4d \neq 4 we have:

every open subset of d\mathbb{R}^d which is homeomorphic to 𝔹 d\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 4\mathbb{R}^4.

The category of Cartesian spaces

See CartSp.

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

References

  • 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 CartSpCartSp 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 ThCartSpThCartSp of infinitesimally thickened Cartesian spaces is known as the site for the Cahiers topos. It is considered

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

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

Revised on July 19, 2014 09:06:10 by Todd Trimble (67.81.95.215)