Urysohn universal space
Encyclopedia
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
.
and has the following property:
s with the supremum norm admits isometric embeddings of all separable metric spaces ("Fréchet
embedding"), as does the space C[0,1] of all continuous function
s [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 "homogenity" 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.
.
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
that contains all separable metric spaces in a particularly nice manner. This mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
concept is due to Pavel Samuilovich Urysohn
Pavel Samuilovich Urysohn
Pavel Samuilovich Urysohn, Pavel Uryson was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology...
.
Definition
A metric space (U,d) is called Urysohn universal if it is separable and completeComplete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
and has the following property:
- given any finite metric space X, any point x in X, and any isometric embeddingIsometryIn mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
f : X\{x} → U, there exists an isometric embedding F : X → U that extends f, i.e. such that F(y) = f(y) for all y in X\{x}.
Properties
If U is Urysohn universal and X is any separable metric space, then there exists an isometric embedding f:X → U. (Other spaces share this property: for instance, the space l∞ of all bounded real sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s with the supremum norm admits isometric embeddings of all separable metric spaces ("Fréchet
Maurice René Fréchet
Maurice Fréchet was a French mathematician. He made major contributions to the topology of point sets and introduced the entire concept of metric spaces. He also made several important contributions to the field of statistics and probability, as well as calculus...
embedding"), as does the space C[0,1] of all continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s [0,1]→R, again with the supremum norm, a result due to Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
.)
Furthermore, every isometry between finite subsets of U extends to an isometry of U onto itself. This kind of "homogenity" 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
Urysohn proved that an Urysohn universal space exists, and that any two Urysohn universal spaces are isometricIsometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
.