Let
Q
=
[
−
1
,
1
]
ω
Q = {[ - 1,1]^\omega }
be the Hilbert cube and
\[
Q
f
=
{
(
x
i
)
∈
Q
|
x
i
=
0
except for finitely many
i
}
.
{Q_f} = \left \{ {({x_i}) \in Q|{x_i} = 0{\text {except for finitely many }}i} \right \}.
\]
For a compact connected polyhedron
X
X
with
dim
X
>
0
\dim X > 0
, the hyperspaces of (nonempty) subcompacta, subcontinua, and compact subpolyhedra of
X
X
are denoted by
2
X
,
C
(
X
)
{2^X},C(X)
, and
Pol(
X
)
{\text {Pol(}}X{\text {)}}
, respectively. And let
C
Pol
(
X
)
=
C
(
X
)
∩
Pol(
X
)
{C^{{\text {Pol}}}}(X) = C(X) \cap {\text {Pol(}}X{\text {)}}
. It is shown that the pair
(
2
X
,
Pol(
X
)
)
({2^X},{\text {Pol(}}X{\text {)}})
is homeomorphic to
(
Q
,
Q
f
)
(Q,{Q_f})
. In case
X
X
has no free arc, it is also proved that the pair
(
C
(
X
)
,
C
Pol
(
X
)
)
(C(X),{C^{{\text {Pol}}}}(X))
is homeomorphic to
(
Q
,
Q
f
)
(Q,{Q_f})
.