A Metric on a Cartesian Product of Metric Spaces
Let
and
represent metric spaces. We can define a metric space
on the cartesian product
by
for
and
We test
and
in turn.
M1:
and
iff
and
iff
iff
so M1 is satisfied.
M2:
so M2 is satisfied.
M3:
so M3 is satisfied andis a metric space.
Other possible metric spaces for a cartesian product exist – the discrete metric
and the ordinary cartesian distance metric are examples.
