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

By S. M. Srivastava (auth.)

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A topological space whose topology is induced by a metric is called a metrizable space. 3) consists of all subsets of X. We call this topology the discrete topology on X. 1 induce the same topology. This topology is called the usual topology. Another such example is obtained as follows. Let d be a metric on X and p(x,y) = min{d(x,y), I}, x,y EX. Then both d and p induce the same topology on X. These examples show that a topology may be induced by more than one metric. Two metrics d and p on a set are called equivalent if they induce the same topology.

The metric d1 will be referred to usual metric on Rn. ). Define 2n1+1 min{/x n -fln/, 1}. n Then d is a metric on RN. 3 If X is any set and d( ) x, 1/ = {O if x = 1/, 1 otherwise, then d defines a metric on X, called the discrete metric. 4 Let (Xo,do), (Xl,dt), (X2,d2), ... be metric spaces and X = TInXn. Fix x = (xo,X1l"') and 1/ = (1/0,1/1, ... ) in X. Define d(x,1/) = L 2n~1 min{dn(xn,1/n),1}. n Then d is a metric on X, which we shall call the product metric. Note that if (X, d) is a metric space and Y ~ X, then d resticted to Y (in fact to Y x Y) is itself a metric.

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. It is easily seen that limx E Fn. Let x :F y. Then d(x, y) > O. Since diameter(Fn ) - 0, there is an integer n such that both x and y cannot belong to Fn.

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