By Czes Kosniowski

ISBN-10: 0521298644

ISBN-13: 9780521298643

This self-contained advent to algebraic topology is appropriate for a few topology classes. It involves approximately one region 'general topology' (without its ordinary pathologies) and 3 quarters 'algebraic topology' (centred round the primary staff, a with no trouble grasped subject which provides a good suggestion of what algebraic topology is). The publication has emerged from classes given on the collage of Newcastle-upon-Tyne to senior undergraduates and starting postgraduates. it's been written at a degree in an effort to let the reader to take advantage of it for self-study in addition to a direction ebook. The process is leisurely and a geometrical flavour is obvious all through. the numerous illustrations and over 350 workouts will turn out necessary as a educating reduction. This account can be welcomed by means of complex scholars of natural arithmetic at schools and universities.

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**Example text**

5 Quotient topology (and groups acting on spaces) In the last chapter we essentially considered a set S, a topological space X and an injective mapping from S to X. This gave us a topology on S: the induced topology. In this chapter we shall consider a topological space X, a set Y and a sul)ective mapping from X to Y. This will give us a topology on Y: the so-called 'quotient' topology. 1 Suppose that f: X -÷ Y is a surjective mapping from a topological space X onto a set Y. The quotient topology on Y with respect to f is the family {U;f'(U)isopeninX}.

Show that this defines an action of H on G. (c) Let G be a group and let,Y(G) denote the set of subsets of G. Show that { ghhEU} ,gEG,UE5°(G) defines an action of G on5'°(G). (d) Let G act on X and define the stabilizer of x E X to be the set {gEG;gx=x}. Prove that is a subgroup of G. (e) Let G act on X and define the orbit of x E X to be the subset 1 gx;gEG } of X. Prove that two orbits Gx, are either disjoint or equal. Deduce that a Gset X decomposes into a union of disjoint subsets. An important consequence of the definition of a G-set X is that in fact G acts on X via bijections.

Continuing in this manner we of { j such that no get a sequence of intervals [a1,b1J. , j E I } covers any of the intervals. Furthermore finite subcollection of f = 2-n and < for all n. } . } . } Let b be the greatest lower bound of the set { b1 ,b2 ,... } By definition we - - wehaveb- a< for each n and so a = b. Since f El. j E I } covers (0,11 and a b E [0,11 we have a E U3 for for is open there is an open interval (a-e,a+e) c a positive integer N so that 2 and hence bN - aN (c. However a E [aN,bN] and a - aN

### A First Course in Algebraic Topology by Czes Kosniowski

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