Topological Manifolds

The main purpose of this chapter is to introduce our basic arena of study, the topological manifold, which we take to be a connected, Hausdorff topological space which is locally like euclidean space $${\mathbb R}^n$$ R n . We present some examples and some standard topological properties enjoyed by all manifolds, such as the Tychonoff property and path connectedness. We also show that manifolds have cardinality $$\mathfrak c$$ c . The simplest examples of non-metrisable manifolds are the open long ray and the long line, and we define them and investigate their properties, especially that any bounded interval in either is homeomorphic to an interval in the real line and, what always surprises beginners, the fact that any continuous function from the open long ray to the real line is eventually constant. Some standard constructions of non-metrisable manifolds are presented, including versions of the Prüfer manifold, Moore’s way of identifying two boundary components to eliminate them as boundary components and Nyikos’s method of inserting a closed long ray into the open unit square of the real plane.