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

## TL;DR

This paper introduces proximal Cech nerves and complexes for approximating the shapes of irregular finite planar regions, offering a simpler alternative to traditional methods and extending the Edelsbrunner-Harer Nerve Theorem.

## Contribution

It presents a novel approach using proximal Cech nerves and complexes to efficiently approximate shapes of planar regions, extending existing nerve theorems.

## Key findings

- Cech nerves are proximal and easier to construct than Alexandroff nerves.
- Cech complexes effectively cover irregular finite planar regions.
- Extended Edelsbrunner-Harer Nerve Theorem for Cech nerves.

## 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 $K$.