gtnotes - General Topology Jesper M Mller Matematisk...

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General Topology Jesper M. Møller Matematisk Institut, Universitetsparken 5, DK–2100 København E-mail address : [email protected] URL : http://www.math.ku.dk/ ~ moller
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Contents Chapter 0. Introduction 5 Chapter 1. Sets and maps 7 1. Sets, functions and relations 7 2. The integers and the real numbers 11 3. Products and coproducts 13 4. Finite and infinite sets 14 5. Countable and uncountable sets 16 6. Well-ordered sets 18 7. Partially ordered sets, The Maximum Principle and Zorn’s lemma 19 Chapter 2. Topological spaces and continuous maps 23 1. Topological spaces 23 2. Order topologies 25 3. The product topology 25 4. The subspace topology 26 5. Closed sets and limit points 29 6. Continuous functions 32 7. The quotient topology 36 8. Metric topologies 42 9. Connected spaces 45 10. Compact spaces 51 11. Locally compact spaces and the Alexandroff compactification 57 Chapter 3. Regular and normal spaces 61 1. Countability Axioms 61 2. Separation Axioms 62 3. Normal spaces 64 4. Second countable regular spaces and the Urysohn metrization theorem 66 5. Completely regular spaces and the Stone– ˇ Cech compactification 69 6. Manifolds 71 Chapter 4. Relations between topological spaces 73 Bibliography 75 3
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CHAPTER 0 Introduction These notes are intended as an to introduction general topology. They should be sufficient for further studies in geometry or algebraic topology. Comments from readers are welcome. Thanks to Michal Jablonowski and Antonio D´ ıaz Ramos for pointing out misprinst and errors in earlier versions of these notes. 5
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CHAPTER 1 Sets and maps This chapter is concerned with set theory which is the basis of all mathematics. Maybe it even can be said that mathematics is the science of sets. We really don’t know what a set is but neither do the biologists know what life is and that doesn’t stop them from investigating it. 1. Sets, functions and relations 1.1. Sets. A set is a collection of mathematical objects. We write a S if the set S contains the object a . 1.1. Example . The natural numbers 1 , 2 , 3 , . . . can be collected to form the set Z + = { 1 , 2 , 3 , . . . } . This na¨ ıve form of set theory unfortunately leads to paradoxes. Russel’s paradox 1 concerns the formula S 6∈ S . First note that it may well happen that a set is a member of itself. The set of all infinite sets is an example. The Russel set R = { S | S 6∈ S } is the set of all sets that are not a member of itself. Is R R or is R 6∈ R ? How can we remove this contradiction? 1.2. Definition . The universe of mathematical objects is stratified. Level 0 of the universe consists of (possibly) some atomic objects. Level i > 0 consists of collections of objects from lower levels. A set is a mathematical object that is not atomic. No object of the universe can satisfy S S for atoms do not have elements and a set and an element from that set can not be in the same level. Thus R consists of everything in the universe. Since the elements of R occupy all levels of the universe there is no level left for R to be in. Therefore R is outside the universe, R is not a set. The contradiction has evaporated!
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