By S. M. Srivastava (auth.)

ISBN-10: 3642854737

ISBN-13: 9783642854736

ISBN-10: 3642854753

ISBN-13: 9783642854750

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**Extra resources for A Course on Borel Sets**

**Example text**

Is a descending sequence in W, then the set A = {w n : n ~ O} has no least element. • Let WI and W 2 be two well-ordered sets. If there is an order-preserving bijection 1 : WI - - W 2, then we call WI and W 2 order isomorphic or simply isomorphic. Such a map 1 is called an order isomorphism. If two well-ordered sets Wit W 2 are order isomorphic, we write WI '" W 2. Note that if WI and W2 are isomorphic, they have the same cardinality. 3 Let W = NU{oo}. Let :5 be defined in the usual way on N and let i < 00 for i E N.

Li) For every real number a, {x EX: I(x) < a} is open. (Iii) Whenever a sequence (x n ) in X limsup/(x n ) ~ I(x). 24 Let X be a metric space and Ii : X tinuous maps. Show that the map I : X - IR. defined by JR, i E I, con- I(x) = inf{fi(x) : i E I}, x E X, is upper-semicontinuous. Next we show that the converse of this result is also true. , I(x) ~ g(x) lor all x. Then there is a sequence 01 continuous maps In : X -JR such that I(x) = inf In(x) lor all x. Proof. Let r be any rational number. Set Ur = {x EX: I(x) < r < g(x)}.

Let W be a well-ordered set and W E W. Set W- W(W) = {u E W: u < w}. Sets of the form W (w) are called initial segments of W. 4 Let W be a well-ordered set and U W(u) = u

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

by Thomas

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