Universal space

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

Theorem:{{Cite book|title = Dimension Theory|last1 = Hurewicz|first1 = Witold|series = Princeton Mathematical Series |volume=4 |publisher=Princeton University Press |orig-year = 1941 |year=2015 |isbn =978-1400875665 |pages =56– |chapter=V Covering and Imbedding Theorems §3 Imbedding of a compact n-dimensional space in I_2n+1: Theorem V.2 |chapter-url=https://books.google.com/books?id=_xTWCgAAQBAJ&pg=PA56 |last2 = Wallman|first2 = Henry}}

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

Nöbeling went further and proved:

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

The last theorem was generalized by Lipscomb to the class of metric spaces of [https://www.encyclopediaofmath.org/index.php/Weight_of_a_topological_space weight] $\backslash textstyle\; \backslash alpha$, $\backslash textstyle\; \backslash alpha>\backslash aleph\_\{0\}$: There exist a one-dimensional metric space $\backslash textstyle\; J\_\{\backslash alpha\}$ such that the subspace of $\backslash textstyle\; J\_\{\backslash alpha\}^\{2d+1\}$ consisting of set of points, at most $\backslash textstyle\; d$ of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than $\backslash textstyle\; d$ and whose weight is less than $\backslash textstyle\; \backslash alpha$.{{Cite journal |title = The quest for universal spaces in dimension theory |last = Lipscomb |first = Stephen Leon |date = 2009 |journal = Notices Amer. Math. Soc. |volume=56 |issue=11 |pages=1418–24 |url=https://www.ams.org/notices/200911/rtx091101418p.pdf

}}

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

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

Theorem{{Cite journal |title = Mean dimension, small entropy factors and an embedding theorem. Theorem 5.1 |last = Lindenstrauss |first = Elon |date = 1999 |journal = Inst. Hautes Études Sci. Publ. Math. |volume=89 |issue=1 |pages=227–262 |doi = 10.1007/BF02698858 |s2cid = 2413058 |url = http://www.numdam.org/item/PMIHES_1999__89__227_0/ }}: Let $\backslash textstyle\; d\backslash in\backslash mathbb\{\{N\}\}$. The compact metric topological dynamical system $\backslash textstyle\; (X,T)$ where $\backslash textstyle\; X=([0,1]^\{d\})^\{\backslash mathbb\{\{Z\}\}\}$ and $\backslash textstyle\; T:X\backslash rightarrow\; X$ is the shift homeomorphism

$\backslash textstyle\; (\backslash ldots,x\_\{-2\},x\_\{-1\},\backslash mathbf\{x\_\{0\}\},x\_\{1\},x\_\{2\},\backslash ldots)\backslash rightarrow(\backslash ldots,x\_\{-1\},x\_\{0\},\backslash mathbf\{x\_\{1\}\},x\_\{2\},x\_\{3\},\backslash ldots)$

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

In the same article Lindenstrauss asked what is the largest constant $\backslash textstyle\; c$

such that a compact metric topological dynamical system whose mean dimension is strictly less than $\backslash textstyle\; cd$ and which possesses an infinite minimal factor embeds into $\backslash textstyle\; ([0,1]^\{d\})^\{\backslash mathbb\{\{Z\}\}\}$. The results above implies $\backslash textstyle\; c\; \backslash geq\; \backslash frac\{1\}\{36\}$. The question was answered by Lindenstrauss and Tsukamoto{{Cite journal|last1=Lindenstrauss|first1=Elon|last2=Tsukamoto|first2=Masaki|date=March 2014|title=Mean dimension and an embedding problem: An example|journal=Israel Journal of Mathematics|language=en|volume=199|issue=2|pages=573–584|doi=10.1007/s11856-013-0040-9|doi-access=free|s2cid=2099527|issn=0021-2172}} who showed that $\backslash textstyle\; c\; \backslash leq\; \backslash frac\{1\}\{2\}$ and Gutman and Tsukamoto{{Cite journal|last1=Gutman|first1=Yonatan|last2=Tsukamoto|first2=Masaki|date=2020-07-01|title=Embedding minimal dynamical systems into Hilbert cubes|url=https://doi.org/10.1007/s00222-019-00942-w|journal=Inventiones Mathematicae|language=en|volume=221|issue=1|pages=113–166|doi=10.1007/s00222-019-00942-w|issn=1432-1297|arxiv=1511.01802|bibcode=2020InMat.221..113G|s2cid=119139371}} who showed that $\backslash textstyle\; c\; \backslash geq\; \backslash frac\{1\}\{2\}$. Thus the answer is $\backslash textstyle\; c=\backslash frac\{1\}\{2\}$.

• Universal property
• Urysohn universal space
• Mean dimension

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Category:Mathematical terminology

Category:Topology

Category:Dimension theory

Category:Topological dynamics