Concept in general topology and analysis
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. The category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.
Definition
Throughout,
is a set,
denotes the power set of
and all filters are assumed to be proper/non-degenerate (i.e. a filter may not contain the empty set).
A Cauchy space is a pair
consisting of a set
together a family
of (proper) filters on
having all of the following properties:
- For each
the discrete ultrafilter at
denoted by
is in ![{\displaystyle C.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- If
is a proper filter, and
is a subset of
then ![{\displaystyle G\in C.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- If
and if each member of
intersects each member of
then ![{\displaystyle F\cap G\in C.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
An element of
is called a Cauchy filter, and a map
between Cauchy spaces
and
is Cauchy continuous if
; that is, the image of each Cauchy filter in
is a Cauchy filter base in ![{\displaystyle Y.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Properties and definitions
Any Cauchy space is also a convergence space, where a filter
converges to
if
is Cauchy. In particular, a Cauchy space carries a natural topology.
Examples
- Any uniform space (hence any metric space, topological vector space, or topological group) is a Cauchy space; see Cauchy filter for definitions.
- A lattice-ordered group carries a natural Cauchy structure.
- Any directed set
may be made into a Cauchy space by declaring a filter
to be Cauchy if, given any element
there is an element
such that
is either a singleton or a subset of the tail
Then given any other Cauchy space
the Cauchy-continuous functions from
to
are the same as the Cauchy nets in
indexed by
If
is complete, then such a function may be extended to the completion of
which may be written
the value of the extension at
will be the limit of the net. In the case where
is the set
of natural numbers (so that a Cauchy net indexed by
is the same as a Cauchy sequence), then
receives the same Cauchy structure as the metric space ![{\displaystyle \{1,1/2,1/3,\ldots \}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Category of Cauchy spaces
The natural notion of morphism between Cauchy spaces is that of a Cauchy-continuous function, a concept that had earlier been studied for uniform spaces.
See also
References
- Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989.
- Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.