Geometric Tools

Abstract

This chapter gathers together some useful geometric tools for later reference. The first section presents Morton Brown’s theorem which tells us that if a space is the monotone union of a countable sequence of open subsets each homeomorphic to ${\mathbb R}^n$ then the space itself is homeomorphic to ${\mathbb R}^n$. We then discuss Brown’s Collaring Theorem, which enables us to impose a product structure on a neighbourhood of a metrisable component of the boundary of a manifold. Finally we consider handlebodies, which provide a useful decomposition of a metrisable manifold into simple pieces.