Let
ρ
\rho
be a metric on a space
X
X
and let
s
≥
1
s\geq 1
. The function
ρ
s
(
a
,
b
)
=
ρ
(
a
,
b
)
s
\rho ^s(a,b)=\rho (a,b)^s
is a quasimetric (it need not satisfy the triangle inequality). The function
inf
ρ
s
(
a
,
b
)
\inf \rho ^s(a,b)
defined by the condition
inf
ρ
s
(
a
,
b
)
=
inf
{
∑
0
n
ρ
s
(
z
i
,
z
i
+
1
)
\inf \rho ^s(a,b)=\inf \{\sum _0^n \rho ^s(z_i,z_{i+1})
z
0
=
a
,
z
n
=
b
}
z_0=a, z_n=b\}
is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space
(
X
,
ρ
)
(X,\rho )
. We also give some examples showing how the topology of the space
(
X
,
inf
ρ
s
)
(X,\inf \rho ^s)
can change as
s
s
changes.