The Urysohn universal space is a certain
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
that contains all
separable metric spaces in a particularly nice manner. This
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
concept is due to
Pavel Urysohn
Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are f ...
.
Definition
A metric space (''U'',''d'') is called ''Urysohn universal''
if it is separable and
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and has the following property:
:given any finite metric space ''X'', any point ''x'' in ''X'', and any
isometric embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
''f'' : ''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''\.
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
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s with the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when the ...
admits isometric embeddings of all separable metric spaces ("
Fréchet embedding"), as does the space C
,1of all
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s
,1†’R, again with the supremum norm, a result due to
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
.)
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
Urysohn proved that a Urysohn universal space exists, and that any two Urysohn universal spaces are
isometric. This can be seen as follows. Take
, two Urysohn universal 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 universal space is required to be complete.
References
{{reflist
Metric geometry