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.

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 proved by Nöbeling. Theorem: 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 the 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 .

Universal spaces in topological dynamics

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 equivariant, 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 equivariant continuous mapping. Lindenstrauss proved the following theorem: Theorem : Let \textstyle d\in\mathbb . The compact metric topological dynamical system \textstyle (X,T) where \textstyle X=([0,1]^{d})^{\mathbb } 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 whose mean dimension is strictly less than \textstyle \frac{d}{36} 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 ([0,1]^{d})^{\mathbb } . The results above implies \textstyle c \geq \frac{1}{36} . The question was answered by Lindenstrauss and Tsukamoto who showed that \textstyle c \leq \frac{1}{2} and Gutman and Tsukamoto who showed that \textstyle c \geq \frac{1}{2} . Thus the answer is \textstyle c=\frac{1}{2} .

See also

Universal property Urysohn universal space Mean dimension

References

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