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.
Definition.
Given a class formula_1 of topological spaces, formula_2 is universal for formula_1 if each member of formula_1 embeds in formula_5. Menger stated and proved the case formula_6 of the following theorem. The theorem in full generality was proven by Nöbeling.
Theorem:
The formula_7-dimensional cube formula_8 is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than formula_9.
Nöbeling went further and proved:
Theorem: The subspace of formula_8 consisting of set of points, at most formula_9 of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than formula_9.
The last theorem was generalized by Lipscomb to the class of metric spaces of weight formula_13, formula_14: There exist a one-dimensional metric space formula_15 such that the subspace of formula_16 consisting of set of points, at most formula_9 of whose coordinates are "rational"" (suitably defined), "is universal for the class of metric spaces whose Lebesgue covering dimension is less than formula_9 and whose weight is less than formula_13.
Universal spaces in topological dynamics.
Consider the category of topological dynamical systems formula_20 consisting of a compact metric space formula_21 and a homeomorphism formula_22. The topological dynamical system formula_20 is called minimal if it has no proper non-empty closed formula_24-invariant subsets. It is called infinite if formula_25. A topological dynamical system formula_26 is called a factor of formula_20 if there exists a continuous surjective mapping formula_28 which is eqvuivariant, i.e. formula_29 for all formula_30.
Similarly to the definition above, given a class formula_1 of topological dynamical systems, formula_2 is universal for formula_1 if each member of formula_1 embeds in formula_5 through an eqvuivariant continuous mapping. Lindenstrauss proved the following theorem:
Theorem: Let formula_36. The compact metric topological dynamical system formula_20 where formula_38 and formula_22 is the shift homeomorphism 
formula_40
is universal for the class of compact metric topological dynamical systems whose mean dimension is strictly less than formula_41 and which possess an infinite minimal factor.
In the same article Lindenstrauss asked what is the largest constant formula_42 such that a compact metric topological dynamical system whose mean dimension is strictly less than formula_43 and which possesses an infinite minimal factor embeds into formula_44. The results above implies formula_45. The question was answered by Lindenstrauss and Tsukamoto who showed that formula_46 and Gutman and Tsukamoto who showed that formula_47. Thus the answer is formula_48.