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 $\textstyle {\mathcal {C}}$ of topological spaces, $\textstyle \mathbb {U} \in {\mathcal {C}}$ is universal for $\textstyle {\mathcal {C}}$ if each member of $\textstyle {\mathcal {C}}$ embeds in $\textstyle \mathbb {U}$. 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 $\textstyle [0,1]^{2d+1}$ 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 $\textstyle [0,1]^{2d+1}$ 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 of weight $\textstyle \alpha$, $\textstyle \alpha >\aleph _{0}$: There exist a one-dimensional metric space $\textstyle J_{\alpha }$ such that the subspace of $\textstyle J_{\alpha }^{2d+1}$ 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 $\textstyle {\mathcal {C}}$ of topological dynamical systems, $\textstyle \mathbb {U} \in {\mathcal {C}}$ is universal for $\textstyle {\mathcal {C}}$ if each member of $\textstyle {\mathcal {C}}$ embeds in $\textstyle \mathbb {U}$ through an equivariant continuous mapping. Lindenstrauss proved the following theorem:

Theorem: Let $\textstyle d\in \mathbb {N}$. The compact metric topological dynamical system $\textstyle (X,T)$ where $\textstyle X=([0,1]^{d})^{\mathbb {Z} }$ and $\textstyle T:X\rightarrow X$ is the shift homeomorphism $\textstyle (\ldots ,x_{-2},x_{-1},\mathbf {x_{0}} ,x_{1},x_{2},\ldots )\rightarrow (\ldots ,x_{-1},x_{0},\mathbf {x_{1}} ,x_{2},x_{3},\ldots )$

is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than $\textstyle {\frac {d}{36}}$ 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 $\textstyle ([0,1]^{d})^{\mathbb {Z} }$. The results above implies $\textstyle c\geq {\frac {1}{36}}$. The question was answered by Lindenstrauss and Tsukamoto who showed that $\textstyle c\leq {\frac {1}{2}}$ and Gutman and Tsukamoto who showed that $\textstyle c\geq {\frac {1}{2}}$. Thus the answer is $\textstyle c={\frac {1}{2}}$.

See also

• Universal property
• Urysohn universal space
• Mean dimension