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\section*{Cauchy filter}

\hypertarget{cauchy_filters}{}\section*{{Cauchy filters}}\label{cauchy_filters}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_nonstandard_analysis}{In nonstandard analysis}\dotfill \pageref*{in_nonstandard_analysis} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{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]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

In a [[metric space]], the [[diameter]] of a [[subset]] $A$ is the [[supremum]] of the [[distance]]s $d(x,y)$ 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 $\delta$.

\begin{defn}
\label{metric}\hypertarget{metric}{}
A \textbf{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 $x,y \in A$, $d(x,y) \leq \delta$.

\end{defn}
(It is actually sufficient to consider enough sufficiently small values of $\delta$, say [[rational number|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 $d$.

\begin{defn}
\label{gauge}\hypertarget{gauge}{}
A \textbf{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 $x,y \in A$, $d(x,y) \leq \delta$.

\end{defn}
(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.

\begin{defn}
\label{topgroup}\hypertarget{topgroup}{}
A \textbf{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 $n \in U$, $x n = y$).

\end{defn}
(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.

\begin{defn}
\label{uniform}\hypertarget{uniform}{}
A \textbf{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, $(x,y) \in U$).

\end{defn}
(It is sufficient to consider a base of the uniformity.)

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

\begin{defn}
\label{ucover}\hypertarget{ucover}{}
A \textbf{Cauchy filter} on a uniform space is a [[proper filter]] $F$ with, for each uniform cover $U$, a set $A \in F$ with $A \in U$.

\end{defn}
(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:

\begin{itemize}%
\item for a topological monoid, there is a difference between left and right in $x n = y$ in Definition \ref{topgroup}, giving left-Cauchy and right-Cauchy filters;
\item there is no notion of quasiuniform cover to generalise Definition \ref{ucover}.

\end{itemize}
The most general context is that of a [[Cauchy space]], where the notion of Cauchy filter is axiomatic.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Cauchy filters in all cases above have these properties:

\begin{itemize}%
\item Every Cauchy filter is [[proper filter|proper]].
\item The [[principal ultrafilter]] $U_x$ at any point $x$ is Cauchy.
\item If $F$ is Cauchy, $G$ is [[proper filter|proper]], and $G$ refines $F$ ($G \supseteq F$), then $G$ is Cauchy.
\item If $F$ and $G$ are Cauchy and their [[join]] $F \vee G$ is proper, then their [[intersection]] $F \cap G$ is Cauchy.

\end{itemize}
These conditions form the abstract definition of a [[Cauchy space]].

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

\begin{itemize}%
\item A filter $F$ \textbf{converges} to a point $x$ if $F \cap U_x$ is Cauchy.

\end{itemize}
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 \textbf{[[Cauchy-continuous map|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.)

\hypertarget{in_nonstandard_analysis}{}\subsection*{{In nonstandard analysis}}\label{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 \textbf{[[finite hyperpoint|finite]]} (or \textbf{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).

[[!redirects Cauchy filter]] [[!redirects Cauchy filters]]



\end{document}