Module 1 : General Introduction
Lecture 1 : Introduction
 
   Introduction

General topology, a language for communicating ideas of continuous geometry, provides us useful tools for studying local properties of space. Notions of compactness and connectedness though important, are quite inadequate for obtaining a reasonable understanding of the global geometry of space. For example, the sphere and the torus are not homeomorphic although they are both compact, path-connected, locally connected metric spaces.

Algebraic topology is a powerful tool in global analysis - the study involving the global geometry of space. It is difficult to define precisely at this point what global analysis is. Perhaps the few examples discussed in the following paragraphs may help in understanding this. The most basic example comes from advanced calculus in connection with Stokes' theorem where a student encounters the notion of orientability of a two dimensional surface in $ \mathbb{R}^3$. A sphere is easily seen to be orientable inasmuch as it has ``two sides''. Small pieces of a surface obviously have ``two sides'' but the Möbius band ``has only one side''. How would one formulate a precise notion of an orientable surface and prove that the Möbius band is non-orientable? Is non-orientability an intrinsic property of the surface or does it depend on the way the surface is presented in $ \mathbb{R}^3$?

Frequently one also sees an interplay between local and global analysis. The powerful algebraic techniques that we shall develop streamlines the process of piecing together local information (which is often trivial) to provide non-trivial information on the global geometry of space. A good example illustrating this ``piecing of local information'' is provided by the proof of the famous theorem in complex analysis asserting the impossibility of a continuous branch of the argument function on the punctured plane $ \mathbb{C} - \{0\}$. Although formal use of algebraic topology can be avoided for this specific case, it is less obvious that the function $ \sqrt{1-z^2}$ is holomorphic on $ \mathbb{C} - [-1, 1]$. Analogous problems in several dimensions would be practically intractable without the use of algebraic topology or some other equally powerful tool in global analysis.

The first example in our list is provided by the famous Jordan curve theorem which also arose in connections with complex analysis.