Universal Space

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.

Definition

Given a class $\textstyle \mathcal$ of topological spaces, $\textstyle \mathbb\in\mathcal$ is universal for $\textstyle \mathcal$ if each member of $\textstyle \mathcal$ embeds in $\textstyle \mathbb$. Menger stated and proved the case $\textstyle d=1$ of the following theorem. The theorem in full generality was proven by Nöbeling.

Theorem:
The $\textstyle (2d+1)$-dimensional cube \textstyle ,1 is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than $\textstyle d$.

Nöbeling went further and proved:

Theorem: The subspace of \textstyle ,1 consisting of set of points, at most $\textstyle d$ of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than $\textstyle d$.

The last theorem was generalized by Lipscomb to the class of metric spaces o
weight
$\textstyle \alpha$, $\textstyle \alpha>\aleph_$: There exist a one-dimensional metric space $\textstyle J_$ such that the subspace of $\textstyle J_^$ consisting of set of points, at most $\textstyle d$ of whose coordinates are "rational"'' (suitably defined), ''is universal for the class of metric spaces whose Lebesgue covering dimension is less than $\textstyle d$ and whose weight is less than $\textstyle \alpha$.

Universal spaces in topological dynamics

Consider the category of topological dynamical systems $\textstyle (X,T)$ consisting of a compact metric space $\textstyle X$ and a homeomorphism $\textstyle T:X\rightarrow X$. The topological dynamical system $\textstyle (X,T)$ is called minimal if it has no proper non-empty closed $\textstyle T$-invariant subsets. It is called infinite if $\textstyle , X, =\infty$. A topological dynamical system $\textstyle (Y,S)$ is called a factor of $\textstyle (X,T)$ if there exists a continuous surjective mapping $\textstyle \varphi:X\rightarrow Y$ which is equivariant, i.e. $\textstyle \varphi(Tx)=S\varphi(x)$ for all $\textstyle x\in X$.

Similarly to the definition above, given a class $\textstyle \mathcal$ of topological dynamical systems, $\textstyle \mathbb\in\mathcal$ is universal for $\textstyle \mathcal$ if each member of $\textstyle \mathcal$ embeds in $\textstyle \mathbb$ through an equivariant continuous mapping. Lindenstrauss proved the following theorem:

Theorem: Let $\textstyle d\in\mathbb$. The compact metric topological dynamical system $\textstyle (X,T)$ where \textstyle X=( ,1)^ and $\textstyle T:X\rightarrow X$ is the shift homeomorphism
$\textstyle (\ldots,x_,x_,\mathbf,x_,x_,\ldots)\rightarrow(\ldots,x_,x_,\mathbf,x_,x_,\ldots)$

is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than $\textstyle \frac$ and which possess an infinite minimal factor.

In the same article Lindenstrauss asked what is the largest constant $\textstyle c$ such that a compact metric topological dynamical system whose mean dimension is strictly less than $\textstyle cd$ and which possesses an infinite minimal factor embeds into \textstyle ( ,1)^. The results above implies $\textstyle c \geq \frac$. The question was answered by Lindenstrauss and Tsukamoto who showed that $\textstyle c \leq \frac$ and Gutman and Tsukamoto who showed that $\textstyle c \geq \frac$. Thus the answer is $\textstyle c=\frac$.

See also

* Universal property
* Urysohn universal space
* Mean dimension

References

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Mathematical terminology
Topology
Dimension theory
Topological dynamics