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

By S. M. Srivastava (auth.)

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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.

Show that f : Y --+ X is continuous if and only if 1I'i 0 f is continuous for all i, where 1I'i : X --+ Xi is the projection map. 27 The product of countably many second countable (equivalently separable) metric spaces is second countable. n, Proof. Let X o, Xl! be second countable. Let X = Xi. We show that X has a countable subbase. 8. Let {Uin : n E N} be a base for Xi' Then, by the definition of the product topology, {1I';1(Uin } : i, n E N} is a subbase for X. 8. • A sequence (x n ) in a metric space (X, d) is called a Cauchy sequence if for every f > 0 there is an N E N such that d(xn,x m } < f for all m,n ~ N.

Thus NN is a zero-dimensional Polish space. Note that the product of a family of zero-dimensional spaces is zero-dimensional. A compatible metric and a base for C can be similarly defined. More generally, let A be a discrete space and X = AN be equipped with the product topology. Then X is a zero-dimensional completely metrizable space; it is Polish if and only if A is countable. Let 8 E A

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