Topology

## A Course on Borel Sets by S. M. Srivastava (auth.) PDF By S. M. Srivastava (auth.)

ISBN-10: 3642854737

ISBN-13: 9783642854736

ISBN-10: 3642854753

ISBN-13: 9783642854750

Similar topology books

This can be a softcover reprint of the 1987 English translation of the second one version of Bourbaki's Espaces Vectoriels Topologiques. a lot of the fabric has been rearranged, rewritten, or changed by way of a extra up to date exposition, and a great deal of new fabric has been included during this e-book, reflecting a long time of growth within the box.

The purpose of this e-book is to build different types of areas which comprise all of the C? -manifolds, but also infinitesimal areas and arbitrary functionality areas. To this finish, the thoughts of Grothendieck toposes (and the common sense inherent to them) are defined at a leisurely speed and utilized. by way of discussing themes similar to integration, cohomology and vector bundles within the new context, the adequacy of those new areas for research and geometry might be illustrated and the relationship to the classical method of C?

Additional info for A Course on Borel Sets

Example text

This topology is called the usual topology. Another such example is obtained as follows. Let d be a metric on X and p(x,y) = min{d(x,y), I}, x,y EX. Then both d and p induce the same topology on X. These examples show that a topology may be induced by more than one metric. Two metrics d and p on a set are called equivalent if they induce the same topology. 6 Show that two metrics d and p on a set X are equivalent if and only if for every sequence (x n ) in X and every x E X, d(x n • x) - 0 <==> p(x n • x) - O.

A(k) = 1(/3(k», kEN. For any n, define "Yn by "Yn(m) = r(/3(u(n, m))), mEN. Fix kEN. We have v(a, ("Yn»(k) = = = = This shows that v(a, ("Yn» u(a(k), "YI(k) (r(k))) u(I(/3(k», r(/3(u(l(k), r(k»))) u(I(/3(k», r(/3(k))) /3(k). = /3. • ,8k-1). Let m = I(k) = 1(181}. Put /(J(8) = (1(80),1(81), •• • , 1(8m -1». 13 Idempotence of the Souslin Operation 37 Since i \$ u(i,j) for all i, j, this definition makes sense. De8nltlon of t/J: Let B and m be 88 above and n Since i < n =* u(m, i) < u(m, n) = r(k) = r(IBI}.

5 No well-ordered set W is order isomorphic to an initial segment W(u) 01 itsell. Proof. Let W be a well-ordered set and u E W. Suppose W and W(u) are isomorphic. Let 1 : W - - W(u) be an order isomorphism. For n E N, let Wn = /"(u). Note that Wo = IO(u) = u > II(U) = I(u) = WI. , (w n ) is a descending sequence in W. 2, W is not well-ordered. This contradiction proves our result. 8 Let (Wlt SI) and (W2' S2) be well-ordered sets. Define an order S on WI x W2 as follows. For (WltW2)'(W~,~) E WI X W2, (WI,W2) S (w~,w~) ~ W2 <2 W~ or(W2 = W~ & WI SI wD· Show that S is a well-order on WI x W2.