product of metric spaces

Theorem 1.

Let $\mathrm{(}{X}_i\mathrm{,}{\varrho }_i\mathrm{)}$ be a metric space for each $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}$ wherethe diameter of $X_i$ using ${\varrho }_i$ is less than $\mathrm{1}\mathrm{/}i$. Then the product topology for the space ${\mathrm{\prod }}_{i\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}{X}_i$ is given by the metric

$$\varrho (x,y)=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{2^i}{\varrho }_i(x_i,y_i).$$

Hence, a countable product of metrizable topological spaces is again metrizable.

Since the convergence in the product topology is the pointwise convergence, the same is true for the metric space with the above metric.