Cauchy filters

Idea

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

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

Definitions

In a metric space, the diameter? of a subset $A$ is the supremum of the distances $d(x,y)$ for $x,y \in A$ (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 $\delta$.

Definition

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

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

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

Definition

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

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

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 $F$ with, for each neighbourhood $U$ of the identity, a set $A \in F$ with, for each $x,y \in A$, $x^{-1} y \in U$ (or equivalently, for each $x,y \in A$, for some $n \in U$, $x n = y$).

(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 $U$ to estimate diameter.

Definition

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

(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 $F$ with, for each uniform cover $U$, a set $A \in F$ with $A \in U$.

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

All of the above have non-symmetric versions: quasimetric spaces, quasigauge spaces, topological monoids, 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 $x n = y$ 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 $U_x$ at any point $x$ is Cauchy.
• If $F$ is Cauchy, $G$ is proper, and $G$ refines $F$ ($G \supseteq F$), then $G$ is Cauchy.
• If $F$ and $G$ are Cauchy and their join $F \vee G$ is proper, then their intersection $F \cap G$ 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 $F$ converges to a point $x$ if $F \cap U_x$ 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 $f\colon X \to Y$ between spaces is Cauchy-continuous if, for every Cauchy filter $F$ on $X$, the filter (generated by) $f(F)$ 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 $F$ is Cauchy iff its nonstandard extension $F^*$ contains a hyperset (a collection of hyperpoints) $A$ whose elements are all adequal. So compared to the other definitions, a single $A$ of infinitesimal diameter suffices.

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

A map $f$ between (quasi)uniform spaces is Cauchy-continuous iff its nonstandard extension $f^*$ has the property that $f^*(x)$ and $f^*(y)$ are adequal whenever $x$ and $y$ are adequal and $x$ 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)