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\begin{document}

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\section*{cartesian space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{topological_structure}{Topological structure}\dotfill \pageref*{topological_structure} \linebreak
\noindent\hyperlink{smooth_structures}{Smooth structures}\dotfill \pageref*{smooth_structures} \linebreak
\noindent\hyperlink{the_category_of_cartesian_spaces}{The category of Cartesian spaces}\dotfill \pageref*{the_category_of_cartesian_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{udefn}
A \textbf{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).

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 $\mathbb{R}^n$ is [[Euclidean|Euclidean topology]] the [[real line]] $\mathbb{R}$ with itself where $n$ 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 $\mathbb{R}^n$ with its standard topology (and sometimes smooth structure) is also called \textbf{real $n$-dimensional space} (distinguish from ``real $n$-dimensional vector space'' which is only isomorphic to it as a vector space).

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

\begin{uexample}
In particular:

\begin{itemize}%
\item $\mathbb{R}^0$ is the [[point]],
\item $\mathbb{R}^1$ is the [[real line]],
\item $\mathbb{R}^2$ is the real [[plane]], which may be identified (in two canonical ways) with the [[complex number|complex plane]] $\mathbb{C}$.

\end{itemize}
\end{uexample}
\begin{uremark}
Cartesian spaces carry plenty of further canonical structure:

\begin{itemize}%
\item It is canonically a [[metric space]] and the [[Euclidean topology]] is the corresponding [[metric space|metric topology]].


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


\item A Cartesian space is canonically a[[vector space]] over the [[field]] of [[real number]]s.



\end{itemize}
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 space]]s, 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 $\mathbb{R}^{\aleph_0}$.

\end{uremark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{topological_structure}{}\subsubsection*{{Topological structure}}\label{topological_structure}

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

\begin{displaymath}
\mathbb{B}^n \simeq \mathbb{R}^n
  \,.
\end{displaymath}
\begin{itemize}%
\item [[topological invariance of dimension]]

\end{itemize}
\hypertarget{smooth_structures}{}\subsubsection*{{Smooth structures}}\label{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

\begin{displaymath}
\mathbb{B}^n \simeq \mathbb{R}^n
  \,.
\end{displaymath}
In fact, in $d \neq 4$ there is no choice:

\begin{utheorem}
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]] $\mathbb{R}^n$.

\end{utheorem}
This was shown in (\hyperlink{Stallings}{Stallings}).

\begin{utheorem}
In $d = 4$ the analog of this statement is false. One says that on $\mathbb{R}^4$ there exist [[exotic smooth structure]]s.

\end{utheorem}
\begin{utheorem}
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.

\end{utheorem}
See the first page of (\hyperlink{Ozols}{Ozols}) for a list of references.

\begin{uremark}
In dimension 4 the analog statement fails due to the existence of [[exotic smooth structure]]s on $\mathbb{R}^4$.

\end{uremark}
\hypertarget{the_category_of_cartesian_spaces}{}\subsection*{{The category of Cartesian spaces}}\label{the_category_of_cartesian_spaces}

See [[CartSp]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[super Cartesian space]]


\item [[polydisk]]


\item [[affine space]]



\end{itemize}
In [[complex geometry]] for purposes of [[Cech cohomology]] the role of Cartesian spaces is played by [[Stein manifolds]].

\begin{itemize}%
\item [[topological invariance of dimension]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after [[René Descartes]].

\begin{itemize}%
\item [[John Stallings]], \emph{The piecewise linear structure of Euclidean space} , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. (\href{http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=2118140}{pdf})


\item V. Ozols, \emph{Largest normal neighbourhoods} , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (\href{http://www.jstor.org/stable/2041672}{jstor})



\end{itemize}
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 space]]s. 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

\begin{itemize}%
\item [[Anders Kock]], \emph{Convenient vector spaces embed into the Cahiers topos} (\href{http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0}{numdam})

\end{itemize}
and briefly mentioned in example 2) on p. 191 of

\begin{itemize}%
\item [[Anders Kock]], \emph{Synthetic differential geometry} (\href{http://home.imf.au.dk/kock/sdg99.pdf}{pdf})

\end{itemize}
following the original article

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{Sur les modeles de la geometrie differentielle synthetique} (\href{http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0}{numdam}).

\end{itemize}
With an eye towards [[Frölicher space]]s the site is also considered in section 5 of

\begin{itemize}%
\item Hirokazu Nishimura, \emph{Beyond the Regnant Philosophy of Manifolds} (\href{http://arxiv.org/abs/0912.0827}{arXiv:0912.0827})

\end{itemize}
[[!redirects Cartesian space]] [[!redirects cartesian spaces]] [[!redirects Cartesian spaces]] [[!redirects real n-dimensional space]]



\end{document}