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:^[1] 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 $\textstyle \alpha$.^[2]

Universal spaces in topological dynamics

Let us 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 eqvuivariant, 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 eqvuivariant continuous mapping. Lindenstrauss proved the following theorem:

Theorem^[3]: Let $\textstyle d\in\mathbb{{N}}$. The compact metric topological dynamical system $\textstyle (X,T)$ where $\textstyle X=([0,1]^{36d})^{\mathbb{{Z}}}$ 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 which possess an infinite minimal factor and whose mean dimension is strictly less than $\textstyle d$.

See also

• Universal property
• Urysohn universal space
• Mean dimension

References

1. Hurewicz, Witold; Wallman, Henry (1941). Dimension Theory. Princeton Mathematical Series, v. 4. Princeton University Press, Princeton, N. J.,. pp. Theorem V.2.
2. Lipscomb., Stephen Leon (2009). "The quest for universal spaces in dimension theory.". Notices Amer. Math. Soc. 56 (11): 1418–1424.
3. Lindenstrauss, Elon (1999). "Mean dimension, small entropy factors and an embedding theorem. Theorem 5.1". Inst. Hautes Études Sci. Publ. Math. 89 (1): 227–262.