Filter (mathematics) - Filter On A Set - Filters in Topology - Cauchy Filters

Cauchy Filters

Let be a metric space.

  • To say that a filter base B on X is Cauchy means that for each real number ε>0, there is a B0B such that the metric diameter of B0 is less than ε.
  • Take (xn) to be a sequence in metric space X. (xn) is a Cauchy sequence if and only if the filter base {{xN,xN+1,...} : N ∈ {1,2,3,...} } is Cauchy.

More generally, given a uniform space X, a filter F on X is called Cauchy filter if for every entourage U there is an AF with (x,y) ∈ U for all x,yA. In a metric space this agrees with the previous definition. X is said to be complete if every Cauchy filter converges. Conversely, on a uniform space every convergent filter is a Cauchy filter. Moreover, every cluster point of a Cauchy filter is a limit point.

A compact uniform space is complete: on a compact space each filter has a cluster point, and if the filter is Cauchy, such a cluster point is a limit point. Further, a uniformity is compact if and only if it is complete and totally bounded.

Most generally, a Cauchy space is a set equipped with a class of filters declared to be Cauchy. These are required to have the following properties:

  1. for each x in X, the ultrafilter at x, U(x), is Cauchy.
  2. if F is a Cauchy filter, and F is a subset of a filter G, then G is Cauchy.
  3. if F and G are Cauchy filters and each member of F intersects each member of G, then FG is Cauchy.

The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space.

Read more about this topic:  Filter (mathematics), Filter On A Set, Filters in Topology

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