nLab Cauchy filter

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

Definition

A Cauchy filter on the real numbers is a proper filter $F$ with, for each strictly positive number $\delta \in \mathbb{R}$, a set $A \in F$ with, for each $x,y \in A$, $\vert y - x\vert \leq \delta$.

Sometimes, one uses Cauchy filters on the rational numbers to construct the real numbers in the first place.

Definition

A Cauchy filter on the rational numbers is a proper filter $F$ with, for each strictly positive number $\delta \in \mathbb{Q}$, a set $A \in F$ with, for each $x,y \in A$, $\vert y - x\vert \leq \delta$.

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 decimal fraction $\delta$.)

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 Booij premetric space, we use the premetric to estimate diameter.

Definition

A Cauchy filter on a Booij premetric space is a proper filter $F$ with, for each strictly positive number $\delta$, a set $A \in F$ with, for each $x,y \in A$, $x \sim_\epsilon y$.

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 , giving left-Cauchy and right-Cauchy filters;
• there is no notion of quasiuniform cover to generalise Definition .

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. (Compare that $f$ is uniformly continuous iff $f^*$ has this property regardless of whether $x$ 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).