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 C {\displaystyle \textstyle {\mathcal {C}}} of topological spaces, U ∈ C {\displaystyle \textstyle \mathbb {U} \in {\mathcal {C}}} is universal for C {\displaystyle \textstyle {\mathcal {C}}} if each member of C {\displaystyle \textstyle {\mathcal {C}}} embeds in U {\displaystyle \textstyle \mathbb {U} } . Menger stated and proved the case d = 1 {\displaystyle \textstyle d=1} of the following theorem. The theorem in full generality was proved by Nöbeling. Theorem: The ( 2 d + 1 ) {\displaystyle \textstyle (2d+1)} -dimensional cube [ 0 , 1 ] 2 d + 1 {\displaystyle \textstyle [0,1]^{2d+1}} is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than d {\displaystyle \textstyle d} . Nöbeling went further and proved: Theorem: The subspace of [ 0 , 1 ] 2 d + 1 {\displaystyle \textstyle [0,1]^{2d+1}} consisting of set of points, at most d {\displaystyle \textstyle d} of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than d {\displaystyle \textstyle d} . The last theorem was generalized by Lipscomb to the class of metric spaces of weight α {\displaystyle \textstyle \alpha } , α > ℵ 0 {\displaystyle \textstyle \alpha >\aleph _{0}} : There exist a one-dimensional metric space J α {\displaystyle \textstyle J_{\alpha }} such that the subspace of J α 2 d + 1 {\displaystyle \textstyle J_{\alpha }^{2d+1}} consisting of set of points, at most d {\displaystyle \textstyle d} of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than d {\displaystyle \textstyle d} and whose weight is less than α {\displaystyle \textstyle \alpha } .
Universal spaces in topological dynamics Consider the category of topological dynamical systems ( X , T ) {\displaystyle \textstyle (X,T)} consisting of a compact metric space X {\displaystyle \textstyle X} and a homeomorphism T : X → X {\displaystyle \textstyle T:X\rightarrow X} . The topological dynamical system ( X , T ) {\displaystyle \textstyle (X,T)} is called minimal if it has no proper non-empty closed T {\displaystyle \textstyle T} -invariant subsets. It is called infinite if | X | = ∞ {\displaystyle \textstyle |X|=\infty } . A topological dynamical system ( Y , S ) {\displaystyle \textstyle (Y,S)} is called a factor of ( X , T ) {\displaystyle \textstyle (X,T)} if there exists a continuous surjective mapping φ : X → Y {\displaystyle \textstyle \varphi :X\rightarrow Y} which is equivariant, i.e. φ ( T x ) = S φ ( x ) {\displaystyle \textstyle \varphi (Tx)=S\varphi (x)} for all x ∈ X {\displaystyle \textstyle x\in X} . Similarly to the definition above, given a class C {\displaystyle \textstyle {\mathcal {C}}} of topological dynamical systems, U ∈ C {\displaystyle \textstyle \mathbb {U} \in {\mathcal {C}}} is universal for C {\displaystyle \textstyle {\mathcal {C}}} if each member of C {\displaystyle \textstyle {\mathcal {C}}} embeds in U {\displaystyle \textstyle \mathbb {U} } through an equivariant continuous mapping. Lindenstrauss proved the following theorem: Theorem: Let d ∈ N {\displaystyle \textstyle d\in \mathbb {N} } . The compact metric topological dynamical system ( X , T ) {\displaystyle \textstyle (X,T)} where X = ( [ 0 , 1 ] d ) Z {\displaystyle \textstyle X=([0,1]^{d})^{\mathbb {Z} }} and T : X → X {\displaystyle \textstyle T:X\rightarrow X} is the shift homeomorphism
( … , x − 2 , x − 1 , x 0 , x 1 , x 2 , … ) → ( … , x − 1 , x 0 , x 1 , x 2 , x 3 , … ) {\displaystyle \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 d 36 {\displaystyle \textstyle {\frac {d}{36}}} and which possess an infinite minimal factor. In the same article Lindenstrauss asked what is the largest constant c {\displaystyle \textstyle c} such that a compact metric topological dynamical system whose mean dimension is strictly less than c d {\displaystyle \textstyle cd} and which possesses an infinite minimal factor embeds into ( [ 0 , 1 ] d ) Z {\displaystyle \textstyle ([0,1]^{d})^{\mathbb {Z} }} . The results above implies c ≥ 1 36 {\displaystyle \textstyle c\geq {\frac {1}{36}}} . The question was answered by Lindenstrauss and Tsukamoto who showed that c ≤ 1 2 {\displaystyle \textstyle c\leq {\frac {1}{2}}} and Gutman and Tsukamoto who showed that c ≥ 1 2 {\displaystyle \textstyle c\geq {\frac {1}{2}}} . Thus the answer is c = 1 2 {\displaystyle \textstyle c={\frac {1}{2}}} .
See also Universal property Urysohn universal space Mean dimension
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