By Czes Kosniowski

This self-contained advent to algebraic topology is acceptable for a few topology classes. It contains approximately one sector 'general topology' (without its traditional pathologies) and 3 quarters 'algebraic topology' (centred round the primary crew, a without difficulty 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 with the intention to let the reader to take advantage of it for self-study in addition to a path publication. The procedure is leisurely and a geometrical flavour is obvious all through. the numerous illustrations and over 350 workouts will end up precious as a educating relief. This account may be welcomed by means of complex scholars of natural arithmetic at schools and universities.

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**A First Course in Algebraic Topology**

This self-contained creation to algebraic topology is acceptable for a few topology classes. It includes approximately one zone 'general topology' (without its ordinary pathologies) and 3 quarters 'algebraic topology' (centred round the primary staff, a with ease grasped subject which provides a good suggestion of what algebraic topology is).

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1 } . Denote [½,l J cannot be covered by a finite subcollection of { j by [a,b] one of those intervals, that is [a,bJ cannot be covered by a finite I }. Again at least one of the intervals [a1, subcollection of ( j ½(a1 + b1)] or [½(a1 + b1),b1J cannot be covered by a finite subcollection J } ; denote one such by [a2 ,b2J. 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 following 9' { Define = F(A,B); A is a compact subset of X and B is an open set of Yl. F(X,Y). ) (e) Let X be a compact metrizable topological space. Suppose that Y is * a metric space with metric d, define d on F(X,Y) by d (f,g) = sup d(f(x),g(x)). open topology. (I) A space X is said to be locally compact if for all x E X every neighbourhood of x contains a compact neighbourhood of x. Show that if X is locally compact then the evaluation map e: F(X,Y) X X Y given by e(f,x) = f(x) is continuous.

The second is obtained by sewing together two long strips of paper. 6 have already mentioned that if S is a subspace of X then open sets of S are not necessarily open in X. If, however, S is open in X then open subsets of S are open in X. 4 Lemma (I) If S is open in X then the open sets of S in the induced topology are open in X. Induced topology 25 (ii) If S is closed in X then the closed sets of S in the induced topology are closed in X. P-roof Since the proofs of (i) and (ii) are more or less identical we shall only give the proof of(i).