If (X,d) is a metric space let
d
x
(
y
)
=
d
(
x
,
y
)
{d_x}(y) = d(x,y)
. It is proved that if each x in X has a neighbourhood P with
d
x
(
P
)
{d_x}(P)
not dense in any neighbourhood of 0 in
[
0
,
∞
)
[0,\infty )
then Ind
X
=
0
X = 0
. This metric condition characterizes metrizable spaces which have Čech dimension zero. Three other metric characterizations are given.