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 $I^{2}$.