product of metric spaces

Theorem 1.

Let $\mathrm{(}{X}_i\mathrm{,}{\varrho }_i\mathrm{)}$ be a metric space for each $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}$ where the 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.

Title	product of metric spaces
Canonical name	ProductOfMetricSpaces
Date of creation	2013-03-22 16:11:44
Last modified on	2013-03-22 16:11:44
Owner	kompik (10588)
Last modified by	kompik (10588)
Numerical id	6
Author	kompik (10588)
Entry type	Theorem
Classification	msc 54E35