| Version 17 |
Version 16 |
| Let us consider first the concept of a {\em tree} that enters in the definition of a thin square. |
\subsubsection{Preliminary Data} |
| Thus, a simplified notion of thin square is that of ``a continuous map from the unit square of the real plane into |
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| a Hausdorff space $X_H$ which factors through a tree'' (\cite{BHKP}). |
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| \begin{definition} |
\begin{definition} |
| A {\it tree}, is defined here as the underlying space $ |K| $ of a |
A {\it tree}, is defined here as the underlying space $ |K| $ of a |
| finite $ 1 $-connected $ 1 $-dimensional simplicial complex $ K $ and |
finite $ 1 $-connected $ 1 $-dimensional simplicial complex $ K $ and |
| boundary $ \partial{I}^{2} $ of $ I^{2} = I \times I $ (that is, a \emph{square} (interval) defined here as the Cartesian product of the unit interval $I :=[0,1]$ of real numbers). |
boundary $ \partial{I}^{2} $ of $ I^{2} = I \times I $ (that is, a \emph{square} (interval) defined here as the Cartesian product of the unit interval $I :=[0,1]$ of real numbers). |
| \end{definition} |
\end{definition} |
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| \begin{definition} |
\begin{definition} |
| A \emph{square map} $ u:I^{2} \longrightarrow X $ in a topological space $ X $ is \emph{thin} if there |
A \emph{square map} $ u:I^{2} \longrightarrow X $ in a topological space $ X $ is \emph{thin} if there |
| is a factorisation of $ u $, $$ u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} |
is a factorisation of $ u $, $$ u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} |
| J_{u} \stackrel{p_{u}}{\longrightarrow} X, $$ where $J_{u}$ is a |
J_{u} \stackrel{p_{u}}{\longrightarrow} X, $$ where $J_{u}$ is a |
| \emph{tree} and $ \Phi_{u} $ is \PMlinkname{piecewise linear (PWL)}{GeometricallyAndorAlgebraicallyThinSquares} on the |
\emph{tree} and $ \Phi_{u} $ is \PMlinkname{piecewise linear (PWL)}{GeometricallyAndorAlgebraicallyThinSquares} on the |
| boundary $ \partial{I}^{2} $ of $ I^{2} $. |
boundary $ \partial{I}^{2} $ of $ I^{2} $. |
| \end{definition} |
\end{definition} |
