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dim-prelim.tex
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%%!TEX root =the-dim-prelim.tex
%arXiv
\chapter{Dimension theory}\label{ch:dim}
\section{Definitions}\label{sec:prelim:dim}
In this section, we give definitions of different types of dimension-like invariants of metric spaces and state general relations between them.
The proofs of most of the statements in this section can be found in the book of Witold Hurewicz and Henry Wallman \cite{hurewicz-wallman};
the rest follow directly from the definitions.
\begin{thm}{Hausdorff dimension}
\label{def:HausDim}\index{dimension!Hausdorff dimension}\index{Hausdorff dimension}
Let $\spc{X}$ be a metric space.
Its Hausdorff dimension is defined as
\[\HausDim\spc{X}=\sup\set{\alpha\in\RR}{\HausMes_\alpha(\spc{X})>0},\]
where $\HausMes_\alpha$ denotes the $\alpha$-dimensional Hausdorff measure.
\end{thm}
Let $\spc{X}$ be a metric space and $\{V_\beta\}_{\beta\in\IndexSet[2]}$
be an open cover of $\spc{X}$.
Let us recall two notions in general topology:
\begin{itemize}
\item The \index{order of a cover}\emph{order} of $\{V_\beta\}$ is the supremum of all integers $n$ such that there is a collection of $n+1$ elements of $\{V_\beta\}$ with nonempy intersection.
\item An open cover $\{W_\alpha\}_{\alpha\in\IndexSet}$ of $\spc{X}$ is called a \index{refinement of a cover}\emph{refinement} of $\{V_\beta\}_{\beta\in\IndexSet[2]}$ if for any $\alpha\in\IndexSet$ there is $\beta\in\IndexSet[2]$ such that $W_\alpha\subset V_\beta$.
\end{itemize}
\begin{thm}{Topological dimension}\label{def:TopDim}\index{dimension!topological dimension}\index{topological dimension}
Let $\spc{X}$ be a metric space.
The topological dimension of $\spc{X}$ is defined to be the minimum of nonnegative integers $n$
such that for any open cover of $\spc{X}$ there is a finite open refinement with order~$n$.
If no such $n$ exists, the topological dimension of $\spc{X}$ is infinite.
The topological dimension of $\spc{X}$ will be denoted by $\TopDim\spc{X}$.
\end{thm}
The invariants satisfying the following two statements are commonly called ``dimension'';
for that reason we call these statements axioms.
\begin{thm}{Normalization axiom}
\label{dim-axiom-norm} For any $m\in\ZZ_{\ge0}$,
\[\TopDim\EE^m=\HausDim\EE^m=m.\]
\end{thm}
\begin{thm}{Cover axiom}\label{dim-axiom-sigma}
If $\{A_n\}_{n=1}^\infty$ is a countable closed cover of $\spc{X}$, then
\begin{align*}
\TopDim \spc{X}&=\sup\nolimits_n\{\TopDim A_n\},
\\
\HausDim \spc{X}&=\sup\nolimits_n\{\HausDim A_n\}.
\end{align*}
\end{thm}
\parbf{On product spaces.}
Recall that the direct product $\spc{X}\times\spc{Y}$ of metric spaces $\spc{X}$ and $\spc{Y}$ is defined in Section \ref{sec:metric spaces}. %\ref{direct-products}.
The direct product satisfies the following two inequalities:
\begin{align*}
\TopDim (\spc{X}\times\spc{Y})
&\le
\TopDim \spc{X}+ \TopDim\spc{Y}
\intertext{and}
\HausDim (\spc{X}\times\spc{Y})
&\ge
\HausDim \spc{X}+ \HausDim\spc{Y}.
\end{align*}
These inequalities might be strict.
For topological dimension, strict inequality holds for a pair of Pontryagin surfaces \cite{pontyagin-surface}.
For Hausdorff dimension, an example was constructed by Abram Besicovitch and Pat Moran \cite{besicovitch-moran}.
\medskip
The following theorem follows from \cite[theorems V 8 and VII 2]{hurewicz-wallman}.
\begin{thm}{Szpilrajn's theorem}\label{thm:szpilrajn}
Let $\spc{X}$ be a separable metric space.
Assume $\TopDim\spc{X}\ge m$. Then $\HausMes_m \spc{X}>0$.
In particular,
$\TopDim\spc{X}\le\HausDim\spc{X}$.
\end{thm}
In fact it is true that for any separable metric space $\spc{X}$ we have
\[\TopDim\spc{X}=\inf\{\HausDim\spc{Y}\},\]
where the infimum is taken over all metric spaces $\spc{Y}$ homeomorphic to $\spc{X}$.
\begin{thm}{Definition}
Let $\spc{X}$ be a metric space
and let $F\:\spc{X}\to\RR^m$ be a continuous map.
A point $\bm{z}\in \Im F$ is called a \emph{stable value} of $F$
if there is $\eps>0$ such that $\bm{z}\in\Im F'$
for any \emph{$\eps$-close} to $F$ continuous map $F'\:\spc{X}\to\RR^m$,
that is, $|F'(x)-F(x)|<\eps$ for all $x\in \spc{X}$.
\end{thm}
The next theorem follows from \cite[theorems VI 1$\&$2]{hurewicz-wallman}.
(This theorem also holds for non-separable metric spaces \cite{nagata}, \cite[3.2.10]{engelking}.)
\begin{thm}{Stable value theorem}\label{thm:stable-value}
Let $\spc{X}$ be a separable metric space.
Then $\TopDim\spc{X}\ge m$ if and only if there is a map $F\:\spc{X}\to\RR^{m}$ with a stable value.
\end{thm}
\begin{thm}{Proposition}\label{thm:HausDim+Lip}
Suppose $\spc{X}$ and $\spc{Y}$ are metric spaces
and $\map \:\spc{X}\to \spc{Y}$ satisfies
\[\dist{\map (x)}{\map (x')}{}\ge \eps\cdot\dist[{{}}]{x}{x'}{}\]
for fixed $\eps>0$ and any pair $x,x'\in \spc{X}$.
Then
\[\HausDim \spc{X}\le \HausDim \spc{Y}.\]
In particular, if there is a Lipschitz onto map $\spc{Y}\to \spc{X}$, then
\[\HausDim \spc{X}\le \HausDim \spc{Y}.\]
\end{thm}
\section{Linear dimension}
In addition to $\HausDim$ and $\TopDim$,
we will use the so-called linear dimension}.
It will be applied only to Alexandrov spaces and to their open subsets (in cases both of curvature bounded below and curvature bounded above).
As we shall see, in all these cases $\LinDim$ behaves nicely and is easy to work with.
Recall that a \index{cone map}\emph{cone map} is a map between cones respecting the cone multiplication.
\begin{thm}{Definition of linear dimension}\label{def:lin-dim}\index{dimension!linear dimension}
Let $\spc{X}$ be a metric space with defined angles.
The \index{linear dimension}\emph{linear dimension} of $\spc{X}$ (denoted by $\LinDim\spc{X}$\index{$\LinDim$ (linear dimension)}) is defined as the exact upper bound on $m\in\ZZ_{\ge0}$
such that there is a distance-preserving cone embedding $\EE^m\hookrightarrow \T_p\spc{X}$
for some $p\in \spc{X}$; here $\EE^m$ denotes the $m$-dimensional Euclidean space
and $\T_p\spc{X}$ denotes the tangent space of $\spc{X}$ at $p$ (defined in Section~\ref{sec:tangent-space+directions}).
\end{thm}
Note that $\LinDim$ takes values in $\ZZ_{\ge0}\cup\{\infty\}$.
The linear dimension $\LinDim$ has no immediate relations to $\HausDim$ and $\TopDim$.
Also, $\LinDim$ does not satisfy the cover axiom (\ref{dim-axiom-sigma}).
Note that
\[\LinDim(\spc{X}\times \spc{Y})
=
\LinDim\spc{X}+ \LinDim\spc{Y}
\eqlbl{eq:inverse-product-axiom}\]
for any two metric spaces $\spc{X}$ and $\spc{Y}$ with defined angles.
The following exercise is based on a construction of Thomas Foertsch and Viktor Schroeder \cite{schroeder-foetch};
it shows that the condition on existence of angles in \ref{eq:inverse-product-axiom} cannot be removed.
\begin{thm}{Exercise}\label{ex:schroeder-foetch}
Construct metrics $\rho_1$ and $\rho_2$ on $\RR^{10}$ defined by norms, such that $(\RR^{10},\rho_i)$ do \emph{not} contain an isometric copy of $\EE^2$ but
$(\RR^{10},\rho_1)\times (\RR^{10},\rho_2)$ has an isometric copy of $\EE^{10}$.
\end{thm}
\parbf{Remarks.}
Linear dimension was first introduced by Conrad Plaut \cite{plaut:survey}
under the name \index{local dimension}\index{dimension!local dimension}\emph{local dimension}.
\index{geometric dimension}\index{dimension!geometric dimension}\emph{Geometric dimension}, introduced by Bruce Kleiner \cite{kleiner} is closely related;
it coincides %should we explain why??
with the linear dimension for $\Alex{}$ and $\CAT{}$ spaces.
One can extend the definition to arbitrary metric spaces.
To do this one should modify the definition of tangent space
and take an arbitrary $n$-dimensional Banach space instead of the Euclidean $n$-space.
For Alexandrov spaces (either $\Alex{}$ or $\CAT{}$) this modification is equivalent to our definition.