PlanetPhysics/Thin Square

Let us consider first the [[../PreciseIdea/|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 [[../PiecewiseLinear/|simplicial complex]] $K$ and [[../PiecewiseLinear/|boundary]] $\partial I^2$ of $I^2=I\times I$ (that is, a [[../PiecewiseLinear/|square]] (interval) defined here as the Cartesian product of the unit interval $I:=[0,1]$ of real numbers).

A square map $u:I^2⟶X$ in a [[../CoIntersections/|topological]] space $X$ is thin if there is a factorisation of $u$, $u:I^2\stackrel{{\mathrm{\Phi }}_u}{⟶}{J}_u\stackrel{p_u}{⟶}X,$ where $J_u$ is a tree and ${\mathrm{\Phi }}_u$ is piecewise linear (PWL) on the boundary $\partial I^2$ of $I^2$.