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

By S. M. Srivastava (auth.)

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28 Let (X, d) be a metric space. Show that for any A ~ X, diameter(A) = diameter(cl(A». 29 (Cantor intersection theorem) A metric space (X,d) is complete i/ and only if/or every decreasing sequence Fo ;2 Fl ;2 F2 ~ ... 0/ nonempty closed subsets 0/ X with diameter(Fn) - 0, the intersection Fn is a singleton. nn Proof. Assume that (X,d) is complete. Let (Fn) be a decreasing sequence of nonempty closed sets with diameter converging to O. Choose Xn E Fn. Since diameter(Fn ) - 0, (xn) is Cauchy and so convergent.

A tree on N Thus the empty sequence e belongs to all trees. Elements of T are often called nodes of T. A node u is called terminal if for no a E A, u a E T. A tree T is called finitely splitting if for every node s of T, {a E A : s a E T} is finite. If T is a tree on A, its body is the set A A IT] = {a E AN : Vk(alk E Tn. Thus, members of IT] are the Infinite branches of T. , it has no infinite branch. If [T] ::F 0, we call Till-founded. 1 Show that T is well-founded if and only if there is no sequence (Sn) in T such that· ..

Y is continuous. (ii) Whenever a sequence (Xn) in X converges to a point x, f(x n ) -+ f(x). (iii) For every f > 0, there is a 6 > 0 such that p(f(x), f(y» < f whenever d(x, y) < 6. 44 2. Topological Preliminaries A function / : X - - Y is called a homeomorphism if it is a bijection and both / and /-1 are continuous. A homeomorphism / from X onto a subspace of Y will be called an embedding. It is easy to see that the composition of any two continuous functions (homeomorphisms) is continuous (a homeomorphism).

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