By Sylvie Guerre-Delabriere

ISBN-10: 0585310920

ISBN-13: 9780585310923

ISBN-10: 0824787234

ISBN-13: 9780824787233

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**Sample text**

If n: X u Y-. X u, Y is the identification map, g may be factored through n to give a commutative triangle PROOF xu, Y where k is induced by g. Then k is one to one and onto by the properties of g. To see that k is continuous, let C be closed in W. Then k-'(C) is closed if and only if n-'k-'(C) is closed. But n-'k-'(C) = g-'(C), which is closed since g is continuous. Since X u, Y and W are compact Hausdorff spaces, k is a homeomorphism. 0 Example Consider Sn-' as the boundary of Dn and let h,: Dn- P-l+ Rn be a homeomorphism.

Let f: X + Y and g: Y -+X be maps of topological spaces. If the compositions f o g and g of are each homotopic to the respective identity map, then and g are homotopy inverses of each other. A map f : X -,Y is a homotopy equivalence iff has a homotopy inverse; in this case X and Y are said to have the same homotopy type. 11 PROPOSITION If f: X+ Y is a homotopy equivalence, then f,: H,(X) +Hn(Y) is an isomorphism for each n. 10 f, o g, = (fog), = identity and g, of* = (g of ), = identity so that g, = f;' and f, is an isomorphism.

X,, in X with xo = x , x, = y, and - - (i) xi+l = xi or (ii) xi+'-' xi or (iii) xi -' xi+l - for each i. Then is the equivalence relation generated by -'. It is the least equivalence relation that preserves all of the relations from -'. For example, let X = Sn,n 2 1, and define to be the least equivalence relation on S" for which x - x for all x. The graph of in S" x Sn is the union of the diagonal D and the antidiagonal D' = { ( x , - x ) I x E S"}. This is obviously closed; hence, s"/- is a compact Hausdorff space called real projective n-space, RP(n).

### Classical Sequences in Banach Spaces (Pure and Applied Mathematics) by Sylvie Guerre-Delabriere

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