Equivalent metrics are pairs of metrics 
and 
defined on a set 
that induce the same metric
topology on 
. This is the case iff,
for every point 
of 
and every radius 
, the open ball
 |
(1)
|
contains, for some 
, an open
ball with center 
with respect to 
,
 |
(2)
|
and conversely.
If 
contains at least two points, 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 
and 
,
 |
(4)
|
Another metric 
equivalent to 
is defined by
 |
(5)
|
for all 
. For 
,
 |
(6)
|
and
 |
(7)
|
The metrics 
and 
also have the same Cauchy
sequences. For every 
, 
implies 
, independently of 
and 
, while 
. Since they also have the same limits, 
is a complete metric
space iff 
is a complete metric
space.
In the Euclidean plane 
, the Euclidean metric
 |
(8)
|
has circular open balls. An equivalent, more general metric for positive real
numbers 
and 
is given by
 |
(9)
|
with elliptical open balls. The taxicab metric
 |
(10)
|
has open balls that are interiors of squares. All these metrics are equivalent to the Euclidean metric.
Equivalent metrics need not have the same Cauchy sequences or both define complete metric spaces. For
example, on the real line, 
is equivalent to the Euclidean
metric, where 
is the usual real inverse tangent. The sequence 
is a Cauchy
sequence for 
because 
, but it has no limit
in 
for 
. Thus 
is not a complete
metric space, whereas 
with its Euclidean metric
is a complete metric space (Lebl 2026, p. 279).
