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By Czes Kosniowski

This self-contained advent 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 workforce, a effortlessly grasped subject which supplies a good suggestion of what algebraic topology is). The booklet 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 permit the reader to exploit it for self-study in addition to a path publication. The process is leisurely and a geometrical flavour is obvious all through. the numerous illustrations and over 350 workouts will end up beneficial as a educating reduction. This account can be welcomed by way of complicated scholars of natural arithmetic at schools and universities.

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12 a corollary we get the following result. Corollary If X is a compact Hausdorff G-space with G fInite then X/G is a compact Hausdorff space. et C be a closed subset of X. Then U gC iT gEG where ir: X -+ X/G is the natural projection. Since the action of g E G on X is closed and is a homeomorphism gC is closed for all g G. Thus ii hence ii'(C) is closed which shows that ir is a closed mapping. So, for example, R pa is a compact Hausdorff space. 11 consider a space X with a subset A c X. 13 Corollary If X is a compact Hausdorff space and A is a closed subset of X then X/A is a compact Hausdorff space.

Prove that g is an open mapping if and only if gir is an open mapping. (b) Let X be a G-space with G fInite. Prove that the natural projection ir: X X/G is a closed mapping. (c) Suppose X is a G-space and H is a normal subgroup of G. Show that X/H is a (G/H)-space and that (X/H)/(G/H) X/G. 6 Product spaces Our final general method of constructing new topological spaces from old ones is through the direct product. Recall that the direct product X X Y of two sets X,Y is the set of ordered pairs (x,y) with x E X and y E Y.

Using the above theorem, many of the homeomorphisms in Chapter 5 can now be easily seen. For example the image f(X) of a compact space X In a Hausdorff space under a continuous injective map is homeomorphic to X. We now go on to investigate how the Hausdorff property carries over to subspaces, topological products and quotient spaces. 89 Theorem A subspace S of a Hausdorff space X is Hausdorff. Proof Let x,y be a pair of distinct points in S. Then there are a pair of disand fl S) and fl S) while y is in fl 5).

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