By Czes Kosniowski

ISBN-10: 0521298644

ISBN-13: 9780521298643

This self-contained creation to algebraic topology is appropriate for a few topology classes. It contains approximately one area 'general topology' (without its ordinary pathologies) and 3 quarters 'algebraic topology' (centred round the primary staff, a conveniently grasped subject which supplies a good suggestion of what algebraic topology is). The e-book 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 as a way to allow the reader to take advantage of it for self-study in addition to a path booklet. The strategy is leisurely and a geometrical flavour is clear all through. the various illustrations and over 350 workouts will turn out important as a instructing relief. This account may be welcomed via complex scholars of natural arithmetic at faculties and universities.

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**Example text**

11 is also given. 14 Exercises (a) Let f: X -, Y be a continuous surjective map of a compact space X onto a Hausdorff space Y. Prove that a subset U of Y is open if and is open in X. ) Deduce that Y has the closed if and only if quotient topology determined by f. (b) Prove that the space Y is Hausdorif if and only if the diagonal D = { Y;y1= y2 } in YX Y is a closed subset of YX Y. (c) Let f: X Y be a continuous map. Prove that if Y is Hausdorif then the set { (x1,x2) E X X X; f(x1) = f(x2) } is a closed subset of XXX.

G) Let X be a topological space and let f: X -+ Y be a suijective map. Let denote the quotient topology on Y. Suppose that 'W is a topology on Y so that f: X Y Is continuous with respect to this topology. Prove that if f is a closed or an open mapping then (Y, 'Wf). Furthermore, give examples to show that 1ff is neither open nor closed then (Y,'W) (Y, (h) Suppose that f: X -+ Y is a surjective map from a topological space X to a set Y. Let Y have the quotient topology determined by f and let A be a subspace of X.

C) Let f: X Y be a continuous map. Prove that if Y is Hausdorif then the set { (x1,x2) E X X X; f(x1) = f(x2) } is a closed subset of XXX. (d) Let f: X -+ Y be a map which is continuous, open and onto. Prove that Y is a Hausdorff space if and only if the set (x1,x2) E X X X; f(x1) = f(x2) } is a closed subset of XX X. Let X be a compact Hausdorff space and let Y be a quotient space determined by a map f: X Y. Prove that Y is Hausdorff if and only if f is a closed map. Furthermore, prove that Y is Hausdorff if and only if the set { (x1 ,x2) E X X X; f(x1) = f(x2) } is a closed sub4 (e) setofXX X.

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

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