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By Czes Kosniowski

This self-contained advent to algebraic topology is appropriate for a few topology classes. It comprises approximately one zone 'general topology' (without its ordinary pathologies) and 3 quarters 'algebraic topology' (centred round the basic crew, a with ease grasped subject which supplies a good suggestion of what algebraic topology is). The e-book 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 order to permit the reader to exploit it for self-study in addition to a path ebook. The process is leisurely and a geometrical flavour is obvious all through. the various illustrations and over 350 routines will end up helpful as a instructing reduction. This account could be welcomed through complex scholars of natural arithmetic at schools and universities.

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Then there are a pair of disand fl S) and fl S) while y is in fl 5). Hence S is Hausdorff. In particular every subset of R" with the usual topology is Hausdorff. 10 Theorem Let X and Y be topological spaces. Then X and Y are Hausdorff if and only If X X Y is Hausdorif. Proof Suppose that X and Y are Hausdorff and let w1 = (xj,y1) and w2 be two distinct points of X X Y. If x1 * x2 then we can find two (x2 U1, x2 E U2. The sets U1 X Y and open disjoint sets U1 ,U2 with x1 U2 X Y are disjoint open sets in X X Y with w1 E U1 X Y and w2 E U2 X Y.

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. Proof Let C be a closed subset of X and let p: X X/A denote the natural map. IfC C) A = 0 then p(C)Cisclosed. lfCC)A*Øthenp(C)=p(C-A) Up(C ('I A) which Is closed because C U A.

B) Let F be the topology on R defined by U Fif and only if for each s U there is a t > s such that [s,t) c U. Prove that (R,J) is Hausdorff. (c) Suppose that X and Y are homeomorphic topological spaces. Prove that X is Hausdorff if and only if Y is Hausdorff. The Hausdorff condition is an example of a separation condition. We shall define some of the other separation conditions, but apart from the next few pages we shall only pursue the Hausdorff condition in detail. 3 Definition Let k be one of the integers 0,1,2,3 or 4.

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