In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a universal space is a certain
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
that contains all metric spaces whose
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
is bounded by some fixed constant. A similar definition exists in
topological dynamics In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
Scope
The central object of study in topolog ...
.
Definition
Given a class
of topological spaces,
is universal for
if each member of
embeds in
.
Menger stated and proved the case
of the following theorem. The theorem in full generality was proven by Nöbeling.
Theorem:
The
-dimensional cube
is universal for the class of compact metric spaces whose
Lebesgue covering dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclidean ...
is less than
.
Nöbeling went further and proved:
Theorem: The subspace of
consisting of set of points, at most
of whose coordinates are rational, is universal for the class of
separable metric spaces whose Lebesgue covering dimension is less than
.
The last theorem was generalized by Lipscomb to the class of metric spaces o
weight,
: There exist a one-dimensional metric space
such that the subspace of
consisting of set of points, at most
of whose coordinates are "rational"'' (suitably defined), ''is universal for the class of metric spaces whose Lebesgue covering dimension is less than
and whose weight is less than
.
Universal spaces in topological dynamics
Consider the category of
topological dynamical systems
consisting of a compact metric space
and a homeomorphism
. The topological dynamical system
is called minimal if it has no proper non-empty closed
-invariant subsets. It is called infinite if
. A topological dynamical system
is called a factor of
if there exists a continuous surjective mapping
which is equivariant, i.e.
for all
.
Similarly to the definition above, given a class
of topological dynamical systems,
is universal for
if each member of
embeds in
through an equivariant continuous mapping.
Lindenstrauss proved the following theorem:
Theorem: Let
. The compact metric topological dynamical system
where
and
is the shift homeomorphism
is universal for the class of compact metric topological dynamical systems whose
mean dimension is strictly less than
and which possess an infinite minimal factor.
In the same article Lindenstrauss asked what is the largest constant
such that a compact metric topological dynamical system whose mean dimension is strictly less than
and which possesses an infinite minimal factor embeds into
. The results above implies
. The question was answered by Lindenstrauss and Tsukamoto who showed that
and Gutman and Tsukamoto
who showed that
. Thus the answer is
.
See also
*
Universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
*
Urysohn universal space
*
Mean dimension
References
{{reflist
Mathematical terminology
Topology
Dimension theory
Topological dynamics