PlanetPhysics/Thin Square

Let us consider first the concept of a tree that enters in the definition of a thin square. Thus, a simplified notion of thin square is that of {\em a continuous map from the unit square of the real plane into a Hausdorff space $$X_H$$ which factors through a tree}.

A {\it tree}, is defined here as the underlying space $$ |K| $$ of a finite $$ 1 $$-connected $$ 1 $$-dimensional simplicial complex $$ K $$ and boundary $$ \partial{I}^{2} $$ of $$ I^{2} = I \times I $$ (that is, a square (interval) defined here as the Cartesian product of the unit interval $$I :=[0,1]$$ of real numbers).

A square map $$ u:I^{2} \longrightarrow X $$ in a topological space $$ X $$ is thin  if there is a factorisation of $$ u $$, $$ u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} J_{u} \stackrel{p_{u}}{\longrightarrow} X, $$ where $$J_{u}$$ is a tree and $$ \Phi_{u} $$ is piecewise linear (PWL) on the boundary $$ \partial{I}^{2} $$ of $$ I^{2} $$.