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.
