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

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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.

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

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