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 $\backslash textstyle\; \backslash mathcal$ of topological spaces, $\backslash textstyle\; \backslash mathbb\backslash in\backslash mathcal$ is universal for $\backslash textstyle\; \backslash mathcal$ if each member of $\backslash textstyle\; \backslash mathcal$ embeds in $\backslash textstyle\; \backslash mathbb$. Menger stated and proved the case $\backslash textstyle\; d=1$ of the following theorem. The theorem in full generality was proven by Nöbeling. Theorem: The $\backslash textstyle\; (2d+1)$-dimensional cube $\backslash textstyle,1$ 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 $\backslash textstyle,1$ 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 o

weight

$\backslash textstyle\; \backslash alpha$, $\backslash textstyle\; \backslash alpha>\backslash aleph\_$: There exist a one-dimensional metric space $\backslash textstyle\; J\_$ such that the subspace of $\backslash textstyle\; J\_^$ 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 $\backslash textstyle\; \backslash alpha$.

Universal spaces in topological dynamics

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 eqvuivariant, 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 $\backslash textstyle\; \backslash mathcal$ of topological dynamical systems, $\backslash textstyle\; \backslash mathbb\backslash in\backslash mathcal$ is universal for $\backslash textstyle\; \backslash mathcal$ if each member of $\backslash textstyle\; \backslash mathcal$ embeds in $\backslash textstyle\; \backslash mathbb$ through an eqvuivariant continuous mapping. Lindenstrauss proved the following theorem: Theorem: Let $\backslash textstyle\; d\backslash in\backslash mathbb$. The compact metric topological dynamical system $\backslash textstyle\; (X,T)$ where $\backslash textstyle\; X=(,1)^$ and $\backslash textstyle\; T:X\backslash rightarrow\; X$ is the shift homeomorphism $\backslash textstyle\; (\backslash ldots,x\_,x\_,\backslash mathbf,x\_,x\_,\backslash ldots)\backslash rightarrow(\backslash ldots,x\_,x\_,\backslash mathbf,x\_,x\_,\backslash ldots)$ is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than $\backslash textstyle\; \backslash frac$ 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 $\backslash textstyle\; (,1)^$. The results above implies $\backslash textstyle\; c\; \backslash geq\; \backslash frac$. The question was answered by Lindenstrauss and Tsukamoto who showed that $\backslash textstyle\; c\; \backslash leq\; \backslash frac$ and Gutman and Tsukamoto who showed that $\backslash textstyle\; c\; \backslash geq\; \backslash frac$. Thus the answer is $\backslash textstyle\; c=\backslash frac$.

See also

* Universal property * Urysohn universal space * Mean dimension

References

{{reflist Category:Mathematical terminology Category:Topology Category:Dimension theory Category:Topological dynamics

Definition

Given a class $\backslash textstyle\; \backslash mathcal$ of topological spaces, $\backslash textstyle\; \backslash mathbb\backslash in\backslash mathcal$ is universal for $\backslash textstyle\; \backslash mathcal$ if each member of $\backslash textstyle\; \backslash mathcal$ embeds in $\backslash textstyle\; \backslash mathbb$. Menger stated and proved the case $\backslash textstyle\; d=1$ of the following theorem. The theorem in full generality was proven by Nöbeling. Theorem: The $\backslash textstyle\; (2d+1)$-dimensional cube $\backslash textstyle,1$ 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 $\backslash textstyle,1$ 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 o

weight

$\backslash textstyle\; \backslash alpha$, $\backslash textstyle\; \backslash alpha>\backslash aleph\_$: There exist a one-dimensional metric space $\backslash textstyle\; J\_$ such that the subspace of $\backslash textstyle\; J\_^$ 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 $\backslash textstyle\; \backslash alpha$.

Universal spaces in topological dynamics

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 eqvuivariant, 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 $\backslash textstyle\; \backslash mathcal$ of topological dynamical systems, $\backslash textstyle\; \backslash mathbb\backslash in\backslash mathcal$ is universal for $\backslash textstyle\; \backslash mathcal$ if each member of $\backslash textstyle\; \backslash mathcal$ embeds in $\backslash textstyle\; \backslash mathbb$ through an eqvuivariant continuous mapping. Lindenstrauss proved the following theorem: Theorem: Let $\backslash textstyle\; d\backslash in\backslash mathbb$. The compact metric topological dynamical system $\backslash textstyle\; (X,T)$ where $\backslash textstyle\; X=(,1)^$ and $\backslash textstyle\; T:X\backslash rightarrow\; X$ is the shift homeomorphism $\backslash textstyle\; (\backslash ldots,x\_,x\_,\backslash mathbf,x\_,x\_,\backslash ldots)\backslash rightarrow(\backslash ldots,x\_,x\_,\backslash mathbf,x\_,x\_,\backslash ldots)$ is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than $\backslash textstyle\; \backslash frac$ 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 $\backslash textstyle\; (,1)^$. The results above implies $\backslash textstyle\; c\; \backslash geq\; \backslash frac$. The question was answered by Lindenstrauss and Tsukamoto who showed that $\backslash textstyle\; c\; \backslash leq\; \backslash frac$ and Gutman and Tsukamoto who showed that $\backslash textstyle\; c\; \backslash geq\; \backslash frac$. Thus the answer is $\backslash textstyle\; c=\backslash frac$.

See also

* Universal property * Urysohn universal space * Mean dimension

References

{{reflist Category:Mathematical terminology Category:Topology Category:Dimension theory Category:Topological dynamics