Two metrics 
and 
defined on a space 
are called equivalent if they induce the same metric
topology on 
.
This is the case iff, for every point 
of 
, every ball with center at 
defined with respect to 
:
 |
(1)
|
contains a ball with center 
with respect to 
:
 |
(2)
|
and conversely.
Every metric 
on 
has uncountably many equivalent metrics. For every positive
real number 
,
a "scaled" metric 
can be defined such that for all 
,
 |
(3)
|
In fact, for all 
:
 |
(4)
|
Another metric 
equivalent to 
is defined by
 |
(5)
|
for all 
.
In fact,
 |
(6)
|
and
 |
(7)
|
In the Euclidean plane 
, the metric
 |
(8)
|
with circular balls can be defined in addition to the Euclidean metric. An equivalent more general metric for all positive real numbers 
and 
can be defined as
 |
(9)
|
with elliptic balls, and the taxicab metric
 |
(10)
|
can be defined with square "balls." All these are equivalent to the Euclidean metric.
