Urysohn universal space

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The Urysohn universal space is a certain metric space that contains all separable metric spaces in a particularly nice manner. This mathematics concept is due to Pavel Samuilovich Urysohn.

Definition[edit]

A metric space (U,d) is called Urysohn universal[1] if it is separable and complete and has the following property:

given any finite metric space X, any point x in X, and any isometric embedding f : X\{x} → U, there exists an isometric embedding F : XU that extends f, i.e. such that F(y) = f(y) for all y in X\{x}.

Properties[edit]

If U is Urysohn universal and X is any separable metric space, then there exists an isometric embedding f:XU. (Other spaces share this property: for instance, the space l of all bounded real sequences with the supremum norm admits isometric embeddings of all separable metric spaces ("Fréchet embedding"), as does the space C[0,1] of all continuous functions [0,1]→R, again with the supremum norm, a result due to Stefan Banach.)

Furthermore, every isometry between finite subsets of U extends to an isometry of U onto itself. This kind of "homogeneity" actually characterizes Urysohn universal spaces: A separable complete metric space that contains an isometric image of every separable metric space is Urysohn universal if and only if it is homogeneous in this sense.

Existence and uniqueness[edit]

Urysohn proved that an Urysohn universal space exists, and that any two Urysohn universal spaces are isometric. This can be seen as follows. Take , two Urysohn spaces. These are separable, so fix in the respective spaces countable dense subsets . These must be properly infinite, so by a back-and-forth argument, one can step-wise construct partial isometries whose domain (resp. range) contains (resp. ). The union of these maps defines a partial isometry whose domain resp. range are dense in the respective spaces. And such maps extend (uniquely) to isometries, since a Urysohn space is required to be complete.

References[edit]

  1. ^ Juha Heinonen (January 2003), Geometric embeddings of metric spaces, retrieved 6 January 2009