In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics.
Given a class
C
of topological spaces,
U
∈
C
is universal for
C
if each member of
C
embeds in
U
. Menger stated and proved the case
d
=
1
of the following theorem. The theorem in full generality was proven by Nöbeling.
Theorem: The
(
2
d
+
1
)
-dimensional cube
[
0
,
1
]
2
d
+
1
is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than
d
.
Nöbeling went further and proved:
Theorem: The subspace of
[
0
,
1
]
2
d
+
1
consisting of set of points, at most
d
of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than
d
.
The last theorem was generalized by Lipscomb to the class of metric spaces of weight
α
,
α
>
ℵ
0
: There exist a one-dimensional metric space
J
α
such that the subspace of
J
α
2
d
+
1
consisting of set of points, at most
d
of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than
d
and whose weight is less than
α
.
Let us consider the category of topological dynamical systems
(
X
,
T
)
consisting of a compact metric space
X
and a homeomorphism
T
:
X
→
X
. The topological dynamical system
(
X
,
T
)
is called minimal if it has no proper non-empty closed
T
-invariant subsets. It is called infinite if
|
X
|
=
∞
. A topological dynamical system
(
Y
,
S
)
is called a factor of
(
X
,
T
)
if there exists a continuous surjective mapping
φ
:
X
→
Y
which is eqvuivariant, i.e.
φ
(
T
x
)
=
S
φ
(
x
)
for all
x
∈
X
.
Similarly to the definition above, given a class
C
of topological dynamical systems,
U
∈
C
is universal for
C
if each member of
C
embeds in
U
through an eqvuivariant continuous mapping. Lindenstrauss proved the following theorem:
Theorem: Let
d
∈
N
. The compact metric topological dynamical system
(
X
,
T
)
where
X
=
(
[
0
,
1
]
36
d
)
Z
and
T
:
X
→
X
is the shift homeomorphism
(
…
,
x
−
2
,
x
−
1
,
x
0
,
x
1
,
x
2
,
…
)
→
(
…
,
x
−
1
,
x
0
,
x
1
,
x
2
,
x
3
,
…
)
, is universal for the class of compact metric topological dynamical systems which possess an infinite minimal factor and whose mean dimension is strictly less than
d
.
