Equivalence of Planar Čech Nerves and Complexes

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 main result of this article is extension of the Edelsbrunner–Harer Nerve Theorem for Cech nerves and Cech complexes.