Product of sequence of Polish spaces is Polish

The product of a sequence of Polish spaces is Polish.

A Polish space is a topological space that is separable and completely metrizable.  Let (X_n, \tau_n) be a sequence of Polish spaces.  Let X = \prod X_n with product topology \tau.  Then, since each X_n is completely metrizable, so is X, by p13 (C). Also, since each X_n is separable, so is X, by p3_(A).  Hence, X is Polish.

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