New PDF release: A Comprehensive Introduction To Differential Geometry

By Michael Spivak

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Let 8(X) be the set of all ends of a connected, locally connected, locally compact Hausdorff space X. Define a topology on Xu 8(X) by dlOosing as neighborhoods Nc(co) of an end cO the sets Nc(co) = co(C) U {ends c: c(C) = co(C)}, for all compact C. Show that Xu 8(X) is a compact Hausdorff space. What is� U 8(�), and�n U 8(�n) for n > I? 20. Consider the following three surfaces. l t _ _ _ l�\ r lti r lti r lti r= (a) Surfaces (A) and (C) have one end, while surface (B) does not. (b) Surfaces (A) and (C) are homeomorphic!

For each n there is a subset An of � such that A/n ) consists of one point. 26 Chapter 1 '(d) There are c non-homeomorphic closed totally disconnected subsets of �'. ' . a sequence of points in C. For each sequence n) < nz < . . , one can add a set A i such that its n l1;'h derived set is {cil. (e) There are c non-homeomorphic connected open subsets of �2 25. (a) A manifold-with-boundary could be defined as a metric space M with the property that for each x E M there is a neighborhood U of x and an integer 11 =:: 0 such that U is homeomorphic to an open subset of lHIn.

Projective n-space Il'" is defined as the rol1ection of all sets {p, -p} for p E S". rill see later that the spaces pll for even n differ in a very important way from the same spaces for odd n. One further definition is needed to complete this introduction to manifolds. We have already discussed some spaces which are not manifolds only because they have a "boundary", for example, the Mobius strip and the disc. 1I, but they do have neighborhoods homeomorphic to an important subset of�". \ = {(xt, ...

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A Comprehensive Introduction To Differential Geometry by Michael Spivak

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