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.
Given a class C of topological spaces, U ∈ C is universal for C if each member of C embeds in U . Menger stated and proved the case d = 1 of the following theorem. The theorem in full generality was proven by Nöbeling.
Theorem: The ( 2 d + 1 ) -dimensional cube [ 0 , 1 ] 2 d + 1 is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than d .
Nöbeling went further and proved:
Theorem: The subspace of [ 0 , 1 ] 2 d + 1 consisting of set of points, at most d of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than d .
The last theorem was generalized by Lipscomb to the class of metric spaces of weight α , α > ℵ 0 : There exist a one-dimensional metric space J α such that the subspace of J α 2 d + 1 consisting of set of points, at most d of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than d and whose weight is less than α .
Let us consider the category of topological dynamical systems ( X , T ) consisting of a compact metric space X and a homeomorphism T : X → X . The topological dynamical system ( X , T ) is called minimal if it has no proper non-empty closed T -invariant subsets. It is called infinite if | X | = ∞ . A topological dynamical system ( Y , S ) is called a factor of ( X , T ) if there exists a continuous surjective mapping φ : X → Y which is eqvuivariant, i.e. φ ( T x ) = S φ ( x ) for all x ∈ X .
Similarly to the definition above, given a class C of topological dynamical systems, U ∈ C is universal for C if each member of C embeds in U through an eqvuivariant continuous mapping. Lindenstrauss proved the following theorem:
Theorem: Let d ∈ N . The compact metric topological dynamical system ( X , T ) where X = ( [ 0 , 1 ] 36 d ) Z and T : X → 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 , … ) , is universal for the class of compact metric topological dynamical systems which possess an infinite minimal factor and whose mean dimension is strictly less than d .
