By Czes Kosniowski

ISBN-10: 0521298644

ISBN-13: 9780521298643

This self-contained creation to algebraic topology is appropriate for a couple of topology classes. It involves approximately one area 'general topology' (without its traditional pathologies) and 3 quarters 'algebraic topology' (centred round the primary staff, a easily grasped subject which supplies a good suggestion of what algebraic topology is). The ebook has emerged from classes given on the college of Newcastle-upon-Tyne to senior undergraduates and starting postgraduates. it's been written at a degree to be able to let the reader to take advantage of it for self-study in addition to a direction ebook. The technique is leisurely and a geometrical flavour is obvious all through. the numerous illustrations and over 350 routines will turn out priceless as a instructing reduction. This account might be welcomed through complicated scholars of natural arithmetic at faculties and universities.

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**Example text**

Let Y be where is the equivalence relation on X given by x x' if and only if c A (intuitively Y is X with A shrunk to a point; in general we denote Y = X/'— by X/A). If we give Y the quotient topology with g: X Y being the natural projection then the inverse image of the point E A is A which is not closed in X. Therefore the J E Y where point [xo] is not closed in Y and Y is not Hausdorff. To ensure that a quotient space Y of a Hausdorff space X is Hausdorff we x = x' or { x,x'} need to impose further restrictions on X.

The notion of the topological product of X and Y can be extended to the topological product of a finite number of topological spaces in an obvious way. 2 Exercises X Y2. zable spaces and suppose that they arise from (b) be metrics dx,dy respectively. Show that d defined by d((x1,y1),(x2,y2))max is a metric on X X Y which produces the product space topology on fl, Rm X X Y. Deduce that the product topology on R X with usual topology). is the same as the usual topology on RnxRm. (c) The graph ofafuncuonf: X-+YisthesetofpointsinXX Yofthe form (x,f(x)) for x E X.

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}.

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

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