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Lecture II - Preliminaries from general topology:

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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.