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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Then every nondecreasing family {Up: {J < o} of nonempty open sets is countable. 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.

F)a. 6. A system of sets with A = N It is essential to become familiar with the above notation, as we shall be using it repeatedly while studying set operations on various pointc1asses. Let A be a nonempty set. A family {A. : 8 E A- 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.

It will be convenient to identify a node «bo, eo), (b 1, cd,··· ,(bn -ll Cn-l» of T by (U,11), where u = (bo, bll ••• , bn - 1 ) and 11 = (eo, Cll ... ,Cn-l). Let (U,11), (u', v') be nodes ofT. We write (u,v) --< (u', v') ifu --< u' and v --< v'. The body of T Is identified with e ex e [T] = ((a,~) E BN X eN : Vk«alk, ~Ik) E TH. The meaning of T(u,tI) Is self-explanatory. If T is a tree on B x a E B N , then the section of T at a is defined by T[a] where 11'1 : BN x eN 11'1 ([TD --+ and = {v E e

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

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