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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Proof. Fix a countable base {Vn } for X. Let {J < 0 be such that UP+1 \ Up 1: 0. Let n(fJ) be the first integer m such that VmnUa Clearly, {J --+ 1: 0 & Vm ~ Up+!. • n({J) is one-to-one and the result is proved. 13 Let X be a separable metric space and 0 an ordinal number. Show that every monotone family {Ep : {J < o} of nonempty sets that are all open or all closed is countable. Let X and Y be topological spaces, f : X --+ Y a map, and x EX. We say that f is continuous at x if for every open V containing f(x), there is an open set U containing x such that feU) ~ V.

Let {3 E NN be such that x E Cj3lk for all k. Choose (a, (-y,,» such that v(a, ('Y,,» = (3. Fix m, n and put k = u(m, n). Then Cj3lk = Ba1m,'Yml n by definition. So, x E A by the above series of equivalences. (-Y,,». 36 1. Cardinal and Ordinal Numbers It remains to show that the functions u, stated earlier exist. 11, /(J, and 1/J with the properties The definition of u: Defineu:NxN--+Nby u(m,n) = 2m(2n + 1) -I, m,n E N. Then u is a bijection such that for all m, n, and p, m ~ u(m,n) and u(m, n) < u(m,p» if n < p.

Is called regular if A. ~ At whenever 8 >- t. ) We define In all the interesting cases A is finite or A equals N. When A = N we write A instead of AN and call it the Souslin operation. If A = {O, I}, we write A2 for AA. 1' be a family of subsets of X. Put AA(F} = {AA({A,,}): A. , AA(F} is the family of sets obtained by applying the operation AA on a system of sets in F. , ifF is closed under finite intersections, then AA(F) consists of sets obtained by performing the operation AA on a regular system of sets in F.

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

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