Proof That The Cartesian Product of a Metrizable Space With Itself is Metrizable

Theorem

If a spaceis metrizable, thenis metrizable.

Proof

A spaceis said to be metrizable if a metriccan be defined onsuch that the topology induced byis

Define the topology onin the usual way, so that ifthen

Define the metric onas

The topology induced onisandis metrizable.