Universal spaces in the topology and topological dynamics Source: en.wikipedia.org/wiki/Universal_spaces_in_the_topology_and_topological_dynamics

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.

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

Theorem:^[1] The ${\displaystyle \textstyle (2d+1)}$-dimensional cube ${\displaystyle \textstyle [0,1]^{2d+1}}$ is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than ${\displaystyle \textstyle d}$.

Nöbeling went further and proved:

Theorem: The subspace of ${\displaystyle \textstyle [0,1]^{2d+1}}$ consisting of set of points, at most ${\displaystyle \textstyle d}$ of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than ${\displaystyle \textstyle d}$.

The last theorem was generalized by Lipscomb to the class of metric spaces of weight ${\displaystyle \textstyle \alpha }$, ${\displaystyle \textstyle \alpha >\aleph _{0}}$: There exist a one-dimensional metric space ${\displaystyle \textstyle J_{\alpha }}$ such that the subspace of ${\displaystyle \textstyle J_{\alpha }^{2d+1}}$ consisting of set of points, at most ${\displaystyle \textstyle d}$ of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than ${\displaystyle \textstyle d}$ and whose weight is less than ${\displaystyle \textstyle \alpha }$.^[2]

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

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

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

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

In the same article Lindenstrauss asked what is the largest constant ${\displaystyle \textstyle c}$ such that a compact metric topological dynamical system whose mean dimension is strictly less than ${\displaystyle \textstyle cd}$ and which possesses an infinite minimal factor embeds into ${\displaystyle \textstyle ([0,1]^{d})^{\mathbb {Z} }}$. The results above implies ${\displaystyle \textstyle c\geq {\frac {1}{36}}}$. The question was answered by Lindenstrauss and Tsukamoto^[4] who showed that ${\displaystyle \textstyle c\leq {\frac {1}{2}}}$ and Gutman and Tsukamoto^[5] who showed that ${\displaystyle \textstyle c\geq {\frac {1}{2}}}$. Thus the answer is ${\displaystyle \textstyle c={\frac {1}{2}}}$.

• Universal property
• Urysohn universal space
• Mean dimension