An introduction to topology and homotopy by Allan J. Sieradski

By Allan J. Sieradski

This article is an advent to topology and homotopy. themes are built-in right into a coherent entire and built slowly so scholars usually are not beaten. the 1st 1/2 the textual content treats the topology of entire metric areas, together with their hyperspaces of sequentially compact subspaces. the second one half the textual content develops the homotopy classification. there are lots of examples and over 900 routines, representing a variety of trouble. This ebook might be of curiosity to undergraduates and researchers in arithmetic.

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EXERCISES X. 1. If [GJ is any collection of connected sets with nGE[G] G ¥ 0, then UGE[G] G is connected. 2. If M is a connected set, so also is any set M 0 such that M c Moe CI(M). In particular, CI(M) is connected. 34 35 Exercises X 3. A set M is connected if and only if no proper (nonempty) subset of M is both open and closed in M. 4. If N is a connected subset of a connected set M and M - N = M 1 M 2 being separated sets, then M 1 U Nand M 2 U N are connected. U M 2, M 1 and 5. Any component of a set K is closed relative to K.

Let e > 0 be given, and let x, y E CI(M). Then there exist points x', y' E M such that p(x,x') < e/2 and p( y,y') < e/2. Then p(x,y) :-::; p(x,x') + p(x',y') + p(y',y):-::; e + diam M Since x and y were arbitrary, as was e, we have diam CI(M) :-::; diam M. Now suppose e > 0 is given and M = U~ M j, where each Mj is connected and has diameter less than e. Then Mo = U~ C1(M;) (\ Mo, where CI(M;) (\ Mo is connected for each i (since M j c CI(M;) (\ Mo c CI(M j» and diam C1(M;) (\ Mo is less than e.

Let {R •• } be the collection of those R/s which intersect A. For each R •• , we pick ai E R •• n A and let Ao = raJ We claim that Ao is a countable dense set. Choose x E A such that x ¢: Ao, and consider any open set V containing x. Then x E R•. c V for some i, which implies a i E V. Since V was arbitrary, x must be a limit point of Ao. Therefore Ao is a countable dense set, and A is separable. 5. Every compact metric space is separable. Let X be a compact metric space. Consider the collection {V,(x): x E X}, which is an open cover for X.

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