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.
Menger stated and proved the case
The theorem in full generality was proven by Nöbeling.
is universal for the class of compact metric spaces whose Lebesgue covering dimension 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 of weight
: There exist a one-dimensional metric space
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
[2] Consider the category of topological dynamical systems
consisting of a compact metric space
The topological dynamical system
is called minimal if it has no proper non-empty closed
It is called infinite if
A topological dynamical system
if there exists a continuous surjective mapping
Similarly to the definition above, given a class
of topological dynamical systems,
through an equivariant continuous mapping.
The compact metric topological dynamical system
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 question was answered by Lindenstrauss and Tsukamoto[4] who showed that
and Gutman and Tsukamoto[5] who showed that