For
f
f
a nonconstant meromorphic function on
Δ
=
{
|
z
|
>
1
}
\Delta = \{ |z| > 1\}
and
r
∈
(
inf
|
f
|
,
sup
|
f
|
)
r \in (\inf |f|,\sup |f|)
, let
L
(
f
,
r
)
=
{
z
∈
Δ
:
|
f
(
z
)
|
=
r
}
\mathcal {L}(f,r) = \{ z \in \Delta :|f(z)| = r\}
. In this paper, we study the components of
Δ
∖
L
(
f
,
r
)
\Delta \backslash \mathcal {L}(f,r)
along with the level sets
L
(
f
,
r
)
\mathcal {L}(f,r)
. Our results include the following: If
f
f
is an outer function and
Ω
\Omega
a component of
Δ
∖
L
(
f
,
r
)
\Delta \backslash \mathcal {L}(f,r)
, then
Ω
\Omega
is a simply-connected Jordan region for which
(
fr
Ω
)
∩
{
|
z
|
=
1
}
({\text {fr}}\;\Omega ) \cap \{ |z| = 1\}
has positive measure. If
f
f
and
g
g
are inner functions with
L
(
f
,
r
)
=
L
(
g
,
s
)
\mathcal {L}\,(f,r) = \mathcal {L}\,(g,s)
, then
g
=
η
f
α
g = \eta {f^\alpha }
, where
|
η
|
=
1
|\eta | = 1
and
α
>
0
\alpha > 0
. When
g
g
is an arbitrary meromorphic function, the equality of two pairs of level sets implies that
g
=
c
f
α
g = c{f^\alpha }
, where
c
≠
0
c \ne 0
and
α
∈
(
−
∞
,
∞
)
\alpha \in ( - \infty ,\infty )
. In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.