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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 $\textstyle \mathcal$ of topological spaces, $\textstyle \mathbb\in\mathcal$ is universal for $\textstyle \mathcal$ if each member of $\textstyle \mathcal$ embeds in $\textstyle \mathbb$. Menger stated and proved the case $\textstyle d=1$ of the following theorem. The theorem in full generality was proven by Nöbeling. Theorem: The $\textstyle \left(2d+1\right)$-dimensional cube $\textstyle,1$ 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 $\textstyle,1$ 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 o
weight
$\textstyle \alpha$, $\textstyle \alpha>\aleph_$: There exist a one-dimensional metric space $\textstyle J_$ such that the subspace of $\textstyle J_^$ 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 $\textstyle \left(X,T\right)$ consisting of a compact metric space $\textstyle X$ and a homeomorphism $\textstyle T:X\rightarrow X$. The topological dynamical system $\textstyle \left(X,T\right)$ 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 $\textstyle \left(Y,S\right)$ is called a factor of $\textstyle \left(X,T\right)$ if there exists a continuous surjective mapping $\textstyle \varphi:X\rightarrow Y$ which is eqvuivariant, i.e. $\textstyle \varphi\left(Tx\right)=S\varphi\left(x\right)$ for all $\textstyle x\in X$. Similarly to the definition above, given a class $\textstyle \mathcal$ of topological dynamical systems, $\textstyle \mathbb\in\mathcal$ is universal for $\textstyle \mathcal$ if each member of $\textstyle \mathcal$ embeds in $\textstyle \mathbb$ through an eqvuivariant continuous mapping. Lindenstrauss proved the following theorem: Theorem: Let $\textstyle d\in\mathbb$. The compact metric topological dynamical system $\textstyle \left(X,T\right)$ where $\textstyle X=\left(,1\right)^$ and $\textstyle T:X\rightarrow X$ is the shift homeomorphism $\textstyle \left(\ldots,x_,x_,\mathbf,x_,x_,\ldots\right)\rightarrow\left(\ldots,x_,x_,\mathbf,x_,x_,\ldots\right)$ is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than $\textstyle \frac$ 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 $\textstyle \left(,1\right)^$. The results above implies $\textstyle c \geq \frac$. The question was answered by Lindenstrauss and Tsukamoto who showed that $\textstyle c \leq \frac$ and Gutman and Tsukamoto who showed that $\textstyle c \geq \frac$. Thus the answer is $\textstyle c=\frac$.