nLab
Cauchy filter

Cauchy filters

Idea

A Cauchy filter on a space XX is a proper filter on XX that contains sets (meaning subsets of XX) of arbitrarily small diameter?.

The precise definition depends on what sort of space XX is, up to the full generality of a Cauchy space.

Definitions

In a metric space, the diameter? of a subset AA is the supremum of the distances d(x,y)d(x,y) for x,yAx,y \in A (which is a lower real number in general). However, we need not think precisely about these diameters; it is enough to characterise those sets with diameter at most δ\delta.

Definition

A Cauchy filter on a metric space is a proper filter FF with, for each strictly positive number δ\delta, a set AFA \in F with, for each x,yAx,y \in A, d(x,y)δd(x,y) \leq \delta.

(It is actually sufficient to consider enough sufficiently small values of δ\delta, say rational δ\delta or 2 n2^{-n} for nn a natural number.)

In a gauge space, instead of a single number δ\delta to estimate diameter, we use δ\delta together with a gauging distance dd.

Definition

A Cauchy filter on a gauge space is a proper filter FF with, for each gauging distance dd and each strictly positive number δ\delta, a set AFA \in F with, for each x,yAx,y \in A, d(x,y)δd(x,y) \leq \delta.

(It is actually sufficient to consider a base of gauging distances, as well as enough sufficiently small δ\delta.)

In a topological group, we use a neighbourhood of the identity element to estimate diameter.

Definition

A Cauchy filter on a topological group is a proper filter FF with, for each neighbourhood UU of the identity, a set AFA \in F with, for each x,yAx,y \in A, x 1yUx^{-1} y \in U (or equivalently, for each x,yAx,y \in A, for some nUn \in U, xn=yx n = y).

(It is sufficient to consider a neighbourhood base? at the identity. There is no difference between left and right even for nonabelian groups.)

In a uniform space, we use an entourage UU to estimate diameter.

Definition

A Cauchy filter on a uniform space is a proper filter FF with, for each entourage UU, a set AFA \in F with, for each x,yAx,y \in A, x Uyx \approx_U y (that is, (x,y)U(x,y) \in U).

(It is sufficient to consider a base of the uniformity.)

If you want to define uniformities in terms of uniform covers:

Definition

A Cauchy filter on a uniform space is a proper filter FF with, for each uniform cover UU, a set AFA \in F with AUA \in U.

(It is sufficient to consider a base of uniform covers.)

All of the above have non-symmetric versions: quasimetric spaces, quasigauge spaces, topological monoid?s, quasiuniform spaces. The definition of Cauchy filter for these is the same, with these caveats:

  • for a topological monoid, there is a difference between left and right in xn=yx n = y in Definition 3;
  • there is no notion of quasiuniform cover to generalise Definition 5.

The most general context is that of a Cauchy space, where the notion of Cauchy filter is axiomatic.

Properties

Cauchy filters in all cases above have these properties:

  • Every Cauchy filter is proper.
  • The principal ultrafilter U xU_x at any point xx is Cauchy.
  • If FF is Cauchy, GG is proper, and GG refines FF (GFG \supseteq F), then GG is Cauchy.
  • If FF and GG are Cauchy and their join FGF \vee G is proper, then their intersection FGF \cap G is Cauchy.

These conditions form the abstract definition of a Cauchy space.

Furthermore, all of these have a notion of convergence given as follows:

  • A filter FF converges to a point xx if FU xF \cap U_x is Cauchy.

In this way, every Cauchy space becomes a convergence space, which agrees with the usual convergence on metric spaces, uniform spaces, etc.

A function f:XYf\colon X \to Y between spaces is Cauchy-continuous if, for every Cauchy filter FF on XX, the filter (generated by) f(F)f(F) is Cauchy. (These are the morphisms in the category of Cauchy spaces.)

In nonstandard analysis

In nonstandard analysis, the hyperpoints of a (quasi)uniform space have a relation of adequality?. A proper filter FF is Cauchy iff its nonstandard extension F *F^* contains a hyperset (a collection of hyperpoints) AA whose elements are all adequal. So compared to the other definitions, a single AA of infinitesimal diameter suffices.

A hyperpoint xx is finite? (or limited) if there is a proper filter FF (necessarily Cauchy) such that F *F^* contains a hyperset whose elements are all adequal to xx. In the analogy between hyperpoints and ultrafilters, the finite hyperpoints correspond to the Cauchy ultrafilters.

A map ff between (quasi)uniform spaces is Cauchy-continuous iff its nonstandard extension f *f^* has the property that f *(x)f^*(x) and f *(y)f^*(y) are adequal whenever xx and yy are adequal and xx is finite. Presumably one can define a Cauchy space in nonstandard analysis by specifying the finite hyperpoints and the relation of adequality only with these (although perhaps not every Cauchy space arises in this way).

Revised on February 22, 2013 20:29:28 by Toby Bartels (64.89.53.179)