Proof That The Cartesian Product of a Metrizable Space With Itself is Metrizable
Theorem
If a space
is metrizable, then
is metrizable.
Proof
A spaceis said to be metrizable if a metric
can be defined on
such that the topology induced byis
Define the topology on
in the usual way, so that if
then
Define the metric onas
The topology induced onis
andis metrizable.
