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 $$\textstyle d\in\mathbb$$. The compact metric topological dynamical system $$\textstyle (X,T)$$ where $$\textstyle X=([0,1]^{d})^{\mathbb}$$ 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 $$\textstyle ([0,1]^{d})^{\mathbb}$$. 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 $$\textstyle c=\frac{1}{2}$$.