Top - Metric and Topological Spaces T. W. Korner October...

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Unformatted text preview: Metric and Topological Spaces T. W. Korner October 11, 2010 Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). What is presented here contains some results which it would not, in my opinion, be fair to set as book-work although they could well appear as problems. In addition, I have included a small amount of material which appears in other 1B courses. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. These notes are written in L A T E X2 and should be available in tex, ps, pdf and dvi format from my home page Contents 1 Preface 2 2 What is a metric? 3 3 Continuity and open sets for metric spaces 7 4 Closed sets for metric spaces 10 5 Topological spaces 12 6 More on topological structures 14 7 Hausdorff spaces 18 8 Compactness 19 9 Products of compact spaces 24 10 Connectedness 25 1 11 Compactness in metric spaces 28 12 The language of neighbourhoods 30 13 Books 32 14 Exercises 33 15 Some hints 43 16 Some proofs 45 17 Executive summary 76 1 Preface Within the last fifty years the material in this course has been taught at Cambridge in the fourth (postgraduate), third, second and first years or left to students to pick up for themselves. Under present arrangements students may take the course either at the end of their first year (before they have met metric spaces in analysis) or at the end of their second year (after they have met metric spaces). Because of this, the first third of the course presents a rapid overview of metric spaces (either as revision or a first glimpse) to set the scene for the main topic of topological spaces. This arrangement is recognised in the examination structure where the 12 lecture course is treated as though it were an 8 lecture course. The first part of these notes states and discusses the main results of the course. Usually, each statement is followed by directions to a proof in the final part of these notes. Whilst I do not expect the reader to find all the proofs by herself, I do ask that she tries to give a proof herself before looking one up. Some of the more difficult theorems have been provided with hints as well as proofs. In my opinion, the two sections on compactness are the deepest part of the course and the reader who has mastered the proofs of the results therein is well on the way to mastering the whole course. May I repeat that, as I said in the small print, I welcome corrections and comments. 2 2 What is a metric? If I wish to travel from Cambridge to Edinburgh, then I may be interested in one or more of the following numbers....
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