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 $\textstyle d$ .

References

↑ Hurewicz, Witold; Wallman, Henry (1941). Dimension Theory. Princeton Mathematical Series, v. 4. Princeton University Press, Princeton, N. J.,. pp. Theorem V.2.
↑ Lipscomb., Stephen Leon (2009). "The quest for universal spaces in dimension theory". Notices Amer. Math. Soc. 56 (11): 1418–1424.
↑ 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.

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[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.

Universal space

Definition

Universal spaces in topological dynamics

See also

References