In the present chapter we introduce elementary concepts in the theory of maps, vector spaces and topology. A modest knowledge of undergraduate mathematics, such as set theory, calculus, complex analysis and linear algebra is assumed.
The main purpose of this book is to study the application of the theory of manifolds to the problems in physics. Vector spaces and topology are, in a sense, two extreme viewpoints of manifolds. A manifold is a space which locally looks like (or ) but not necessarily globally. As a first approximation, we may model a small part of a manifold by a Euclidean space (or ) (a small area around a point on a surface can be approximated by the tangent plane at that point); this is the viewpoint of a vector space. In topology, however, we study the manifold as a whole. We want to study the properties of manifolds and classify manifolds using some sort of ‘measures’. Topology usually comes with an adjective: algebraic topology, differential topology, combinatorial topology, general topology and so on. These adjectives refer to the measure we use when classifying manifolds.
Problems
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
来源版本：second-edition-local
来源 PDF SHA-256：fcd24c3371d092a8c699ccf5cf0765462924b6c48412804231076e8084b1940d
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