product of metric spaces

Theorem 1.

Let (Xi,ϱi) be a metric space for each i=1,2,, where the diameterPlanetmathPlanetmathPlanetmath of Xi using ϱi is less than 1/i. Then the product topology for the space i=1Xi is given by the metric


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

Since the convergence in the product topology is the pointwise convergenceMathworldPlanetmath, 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