Proximal Planar Cech Nerves. An Approach to Approximating the Shapes of Irregular, Finite, Bounded Planar Regions

Abstract

This article introduces proximal Cech nerves and Cech complexes, restricted to finite, bounded regions $K$ of the Euclidean plane. A Cech nerve is a collection of intersecting balls. A Cech complex is a collection of nerves that cover $K$. Cech nerves are proximal, provided the nerves are close to each other, either spatially or descriptively. A Cech nerve has an advantage over the usual Alexandroff nerve, since we need only identify the center and fixed radius of each ball in a Cech nerve instead of identifying the three vertices of intersecting filled triangles (2-simplexes) in an Alexandroff nerve. As a result, Cech nerves more easily cover $K$ and facilitate approximation of the shapes of irregular finite, bounded planar regions. A main result of this article is an extension of the Edelsbrunner-Harer Nerve Theorem for descriptive and non-descriptive Cech nerves and Cech complexes, covering $K$.