Module 2 : General Topology
Lecture 2 : Preliminaries from general topology
 

We discuss in this lecture a few notions of general topology that are covered in earlier courses but are of frequent use in algebraic topology. We shall prove the existence of Lebesgue number for a covering, introduce the notion of proper maps and discuss in some detail the stereographic projection and Alexandroff's one point compactification. We shall also discuss an important example based on the fact that the sphere $ S^{n}$ is the one point compactifiaction of $ \mathbb{R}^n$. Let us begin by recalling the basic definition of compactness and the statement of the Heine Borel theorem.

Definition 2.1

A space $ X$ is said to be compact if every open cover of $ X$ has a finite sub-cover. If $ X$ is a metric space, this is equivalent to the statement that every sequence has a convergent subsequence.

If $ X$ is a topological space and $ A$ is a subset of $ X$ we say that $ A$ is compact if it is so as a topological space with the subspace topology. This is the same as saying that every covering of $ A$ by open sets in $ X$ admits a finite subcovering. It is clear that a closed subset of a compact subset is necessarily compact. However a compact set need not be closed as can be seen by looking at $ X$ endowed the indiscrete topology, where every subset of $ X$ is compact. However, if $ X$ is a Hausdorff space then compact subsets are necessarily closed. We shall always work with Hausdorff spaces in this course. For subsets of $ \mathbb{R}^n$ we have the following powerful result.