Cauchy space

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.

DefinitionEdit

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:

  1. For each   the discrete ultrafilter at   denoted by   is in  
  2. If     is a proper filter, and   is a subset of   then  
  3. If   and if each member of   intersects each member of   then  

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  

Properties and definitionsEdit

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.

ExamplesEdit

  • 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  

Category of Cauchy spacesEdit

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 alsoEdit

ReferencesEdit

  • 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.