In: Advanced Math
Let q and p be natural numbers, and show that the metric on Rq+p is equivalent to the metric it has if we identify Rq+p with Rq × Rp.
We want to show, the metric topology on
is equivalant to product topology of
. We can identify points of
and
in natural way. Now pick
. Take an open ball arround
of radius
. Now it contains points
, such that
. That is same as saying,
Now,
and
. Take balls
and
of radius
arround
and
. And consider
. Now,
belong to the product topology of
.
contains all points
such that
and
. Then in
,
. So,
.
Next we need to show for
,
is a ball centered at
of radius
, and
is a ball centered at
of radius
, we can find a real number
, such that
. Now, for
,
and for
,
.
With out loss of generality, let
be minimum of
. Then take
. Then
. That imply
and
. So, for this
,
.
So these two topologies are equivalant.