Universal space

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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[edit]

Given a class of topological spaces, is universal for if each member of embeds in . Menger stated and proved the case of the following theorem. The theorem in full generality was proven by Nöbeling.

Theorem:[1] The -dimensional cube is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than .

Nöbeling went further and proved:

Theorem: The subspace of consisting of set of points, at most of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than .

The last theorem was generalized by Lipscomb to the class of metric spaces of weight , : There exist a one-dimensional metric space such that the subspace of consisting of set of points, at most of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than and whose weight is less than .[2]

Universal spaces in topological dynamics[edit]

Let us consider the category of topological dynamical systems consisting of a compact metric space and a homeomorphism . The topological dynamical system is called minimal if it has no proper non-empty closed -invariant subsets. It is called infinite if . A topological dynamical system is called a factor of if there exists a continuous surjective mapping which is eqvuivariant, i.e. for all .

Similarly to the definition above, given a class of topological dynamical systems, is universal for if each member of embeds in through an eqvuivariant continuous mapping. Lindenstrauss proved the following theorem:

Theorem[3]: Let . The compact metric topological dynamical system where and is the shift homeomorphism , 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 .

See also[edit]

References[edit]

  1. ^ Hurewicz, Witold; Wallman, Henry (1941). Dimension Theory. Princeton Mathematical Series, v. 4. Princeton University Press, Princeton, N. J.,. pp. Theorem V.2. 
  2. ^ Lipscomb., Stephen Leon (2009). "The quest for universal spaces in dimension theory.". Notices Amer. Math. Soc. 56 (11): 1418–1424. 
  3. ^ Lindenstrauss, Elon (1999). "Mean dimension, small entropy factors and an embedding theorem. Theorem 5.1". Inst. Hautes Études Sci. Publ. Math. 89 (1): 227–262.