## 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.