By S. M. Srivastava (auth.)

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

**Example text**

For simplicity of notation we shall write s ~ a for s ~ (a). For sEA

We now show that v is onto. Towards this, let /3 E NN. Define a ~ NN by . 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}.

48 2. Topological Preliminaries Two cases arise: g(xo} ~ r or g(xo} > r. If g(xo} ~ r, then Hence, f~(xo} = g(xo} < f(xo} + f Xo E X \ Ur • for all n. If g(xo} > r, then Xo E Ur • Take any n such that Xo E p,;. Then f~(xo} < f(xo} + f, and our result is proved. • We proved the above result under the additional condition that f is dominated by a continuous function. So the question arises; Is every realvalued upper-semicontinuous function defined on a metric space dominated by a continuous function?