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 ${𝒞}$ of topological spaces, $\mathbb{𝕌}\in {𝒞}$ is universal for ${𝒞}$ if each member of ${𝒞}$ embeds in $\mathbb{𝕌}$. Menger stated and proved the case $d=1$ of the following theorem. The theorem in full generality was proven by Nöbeling.

Theorem: The $(2d+1)$-dimensional cube $[0,1{]}^{2d+1}$ 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 $[0,1{]}^{2d+1}$ 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 $\alpha$, $\alpha >{\mathrm{\aleph }}_0$: There exist a one-dimensional metric space $J_{\alpha }$ such that the subspace of $J_{\alpha }^{2d+1}$ 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 $\alpha$.

Universal spaces in topological dynamics

Consider the category of topological dynamical systems $(X,T)$ consisting of a compact metric space $X$ and a homeomorphism $T:X\to X$. 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 $|X|=\mathrm{\infty }$. A topological dynamical system $(Y,S)$ is called a factor of $(X,T)$ if there exists a continuous surjective mapping $\phi :X\to Y$ which is equivariant, i.e. $\phi (Tx)=S\phi (x)$ for all $x\in X$.

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

Theorem: Let $d\in \mathbb{\mathbb{N}}$. The compact metric topological dynamical system $(X,T)$ where $X=([0,1{]}^d{)}^{\mathbb{\mathbb{Z}}}$ and $T:X\to X$ is the shift homeomorphism $(\dots ,x_{-2},x_{-1},{\mathbf{x}}_{\mathbf{𝟎}},x_1,x_2,\dots )\to (\dots ,x_{-1},x_0,{\mathbf{x}}_{\mathbf{𝟏}},x_2,x_3,\dots )$

is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than $\frac{d}{36}$ and which possess an infinite minimal factor.

In the same article Lindenstrauss asked what is the largest constant $c$ such that a compact metric topological dynamical system whose mean dimension is strictly less than $cd$ and which possesses an infinite minimal factor embeds into $([0,1{]}^d{)}^{\mathbb{\mathbb{Z}}}$. The results above implies $c\ge \frac{1}{36}$. The question was answered by Lindenstrauss and Tsukamoto who showed that $c\le \frac{1}{2}$ and Gutman and Tsukamoto who showed that $c\ge \frac{1}{2}$. Thus the answer is $c=\frac{1}{2}$.

See also

• Universal property
• Urysohn universal space
• Mean dimension