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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 (2d+1)-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 (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 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 (X,T) where \textstyle X=(,1)^ and \textstyle T:X\rightarrow X is the shift homeomorphism \textstyle (\ldots,x_,x_,\mathbf,x_,x_,\ldots)\rightarrow(\ldots,x_,x_,\mathbf,x_,x_,\ldots) 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 (,1)^. 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.

See also

* Universal property * Urysohn universal space * Mean dimension

References

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