A General Topology Workbook by Iain Adamson

By Iain Adamson

This booklet has been referred to as a Workbook to make it transparent from the beginning that it isn't a traditional textbook. traditional textbooks continue by way of giving in every one part or bankruptcy first the definitions of the phrases for use, the suggestions they're to paintings with, then a few theorems related to those phrases (complete with proofs) and eventually a few examples and workouts to check the readers' figuring out of the definitions and the theorems. Readers of this booklet will certainly locate all of the traditional constituents--definitions, theorems, proofs, examples and exercises­ yet now not within the traditional association. within the first a part of the publication might be came across a short evaluate of the fundamental definitions of normal topology interspersed with a wide num­ ber of routines, a few of that are additionally defined as theorems. (The use of the observe Theorem isn't meant as a sign of trouble yet of value and usability. ) The workouts are intentionally now not "graded"-after all of the difficulties we meet in mathematical "real life" don't are available order of trouble; a few of them are extremely simple illustrative examples; others are within the nature of educational difficulties for a conven­ tional direction, whereas others are rather tough effects. No suggestions of the workouts, no proofs of the theorems are integrated within the first a part of the book-this is a Workbook and readers are invited to aim their hand at fixing the issues and proving the theorems for themselves.

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Sec. R. We have given a method of deciding if a compact surface is non-orientable by looking at its plane model. Finally we have stated that it is possible using the connected sum construction to build every compact surface imaginable from the four basic surfaces S, T, K and P. 6 COMMENTS It seems reasonable to call the surface K a (Klein) bottle, since it could be considered to be a bottle with no inside and no outside! 2 a a a- 1 c Fig. 9. But why should we call the surface P a projective plane?

Fig. 2. Sec. 11] Exercises 31 (8) (a) Construct paper models for Tand K using the directions in Fig. 1. 2, and the semicircles representing their semi-disc neighbourhoods. Examine the finished models to see how these semi-discs fit together to form a disc neighbourhood of the identified point p. (b) Sketch a sequence of clockwise arrows on your model for K, using Fig. 1 as a guide, to show that K is non-orientable. (c) What do you get when you cut your paper model of K in half along the line I shown in Fig.

6 THE CLASSIFICATION THEOREM Let us consider again our fundamental theorem in the light of our new definitions. 1 The Classification Theorem. (a) An orientable compact surface is homeomorphic to nTfor some n~O. (b) A non-orientable compact surface is homeomorphic to either (n1)K or (n1)P, for some n~O. 1. 1, we saw that any non-orientable compact surface M is homeomorphic to a surface of form: (n1) (mP) for some integers n~O and m~1. The proof now proceeds by induction on m. If m = 1, then M takes one of the forms in (b) for any value of n.

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