An introduction to topology and homotopy by Allan J. Sieradski PDF

By Allan J. Sieradski

ISBN-10: 0534929605

ISBN-13: 9780534929602

This article is an creation to topology and homotopy. subject matters are built-in right into a coherent complete and constructed slowly so scholars aren't beaten. the 1st 1/2 the textual content treats the topology of whole metric areas, together with their hyperspaces of sequentially compact subspaces. the second one 1/2 the textual content develops the homotopy type. there are various examples and over 900 workouts, representing a variety of hassle. This publication could be of curiosity to undergraduates and researchers in arithmetic.

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Example text

Finally, let the a , w e i g h t of X be given by Wc~(X) = sup{w(A): A E a}. 1. Proof. To For each X and each a, wa(X ) < c(Ca(X)) <_ d(Ca(X)) = ww(X). show that d(Ca(X)) = ww(X), since suffices to show t h a t d(Ck(X)) ~ ww(X) _< d(Cp(X)). bijection, where w(Y) = ww(X). Cp(X) _< C a ( X ) -< Ck(X ), it So let ~: X -* Y be a continuous Then the induced function ¢*: Ck(Y ) ~ Ck(X ) is almost onto, so t h a t d(Ck(X)) _< d(Ck(Y)) _< nw(Ck(Y)) <_ w(Y) = ww(X). To see t h a t ww(X) <_ d(Cp(X)), let D be an infinite dense subset of Cp(X) having cardinality d(Cp(X)).

Since f o r a r e t h e same, t h e n t h i s t h e o r e m a l s o c h a r a c t e r i z e s t h e 1 r - c h a r a c t e r o f C a ( X ). 1. Proof. For each X a n d each a, x ( C a ( X ) ) = 7 r x ( C a ( X ) ) = ha(X). L e t {[At,Vt]: t E T} be t h e I TJ _< x ( C a ( X ) ) . Suppose, such that A is n o t c o n t a i n e d in A t [At,Vt] c [A,(-1,1)]. fat) = {0}. But Let then by I TI the < reverse ha(X). Let of f e base a t t E T. let Y be topological the {At: t With there there this f0 exists is a t contradiction, such t h a t an A E a E T so t h a t f(a) then = 1 and it is s e e n --- x ( C a ( X ) ) .

U (Uzn×V zn )" T h e r e f o r e C c ( A z l X B Z l ) U ... U (A zn xBzn) c W. The local compactness insure t h a t E is onto. 7 is only needed to This may also be obtained by taking X x Y as a k - s p a c e , as given by the n e x t corollary. 8. If X×Y is a k - s p a c e , then E: Ck(XxY ) ~ Ck(X,Ck(Y)) is a homeomor p hism. Proof. To show t h a t E is onto, let g E Ck(X,Ck(Y) ). Since X x Y is a k - s p a c e , it suffices to show t h a t E - l ( g ) t A x B is continuous, where A and B are compact subsets of X and Y, respectively.

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An introduction to topology and homotopy by Allan J. Sieradski

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