Filterbases

A subset $F$ of a lattice $L$ is a filterbase if it becomes a filter when closed under axiom (1). Equivalently, a filterbase is any downward-directed subset. Any subset of a meet-semilattice may be used as a filter subbase; form a filterbase by closing under finite meets.

A filterbase $F$ of sets is proper (that is, it generates a proper filter of sets) iff each set in $F$ is inhabited. A filter subbase of sets is proper iff it satisfies the finite intersection property (well known in topology from a criterion for compact spaces): every finite collection from the subfilter has inhabited intersection.

Application to analysis and topology

Every net $\nu: I \to S$ defines an eventuality filter $E_\nu$: let $A$ belong to $E_\nu$ if, for some index $k$, for every $l \geq k$, $\nu_l \in A$. (That is, $\nu$ is eventually in $A$.) Note that $E_\nu$ is proper; conversely, any proper filter $F$ has a net whose eventuality filter is $F$ (as described at net). Everything below can be done for nets as well as filters, but filters often lead to a cleaner theory.

Convergent filters

A convergence space is a set $S$ together with a relation $\to$ from $\mathcal{F}S$ to $S$, where $\mathcal{F}S$ is the set of filters on $S$, satisfying some axioms. (If $F \to x$, we say that $F$ converges to $x$.) The axioms are these: 1. If $F \subseteq G$ and $F \to x$, then $G \to x$; 1. The principal ultrafilter at $x$ converges to $x$; 1. If $F \to x$ and $G \to x$, then $F \cap G \to x$; 1. If $F \to x$ and $A \in F$, then $x \in A$.

In other words, the set of filters that converge to $x$ is a filter of subsets of the set of subfilters of the principal ultrafilter at $x$. (But that is sort of a tongue twister.)

The morphisms of convergence spaces are the continuous functions; a function $f$ between convergence spaces is continuous if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A) \;|\; A \in F\}$. In this way, convergence spaces form a category $Conv$.

Any topological space is a convergence space: $F \to x$ if every neighbourhood of $x$ belongs to $F$. A convergence space is topological if it comes from a topology on $S$. The full subcategory of $Conv$ consisting of the topological convergence spaces is equivalent to the category Top of topological spaces. In this way, the definitions in this section all become theorems about topological spaces. However, not every convergence in analysis is topological; convergence in measure is a good counterexample.

Note that the improper filter (the power set of $S$) converges to every point. Conversely, a convergence space $S$ is Hausdorff if every proper filter converges to at most one point; then we have a partial function $\lim$ from $\mathcal{F}S$ to $S$.

A convergence space $S$ is compact if every proper filter is contained in a convergent filter. Equivalently (assuming the Boolean prime ideal theorem), $S$ is compact iff every ultrafilter converges.

Cauchy filters

A Cauchy space is a set $S$ together with a collection of filters declared to be Cauchy filters. These must satisfy axioms: 1. If $F \subseteq G$, $F$ is Cauchy, and $G$ is proper, then $G$ is Cauchy; 1. Every principal ultrafilter is Cauchy; 1. If $F$ and $G$ are Cauchy and $F \cap G$ is proper, then $F \cap G$ is Cauchy; 1. Every Cauchy filter is proper.

In other words, the set of Cauchy filters is a filter of subsets of the set of proper filters (another tongue twister).

The morphisms of Cauchy spaces are the Cauchy functions; a function $f$ between Cauchy spaces is Cauchy if $f(F)$ is Cauchy whenever $F$ is. In this way, Cauchy spaces form a category $Cauchy$.

Any metric space is a Cauchy space: $F$ is Cauchy if it has elements of arbitrarily small diameter. (In particular, a sequence is a Cauchy sequence iff its eventuality filter is Cauchy.) In this way, the category of metric spaces becomes a subcategory (but not a full one) of $Cauchy$. (This can be generalised to uniform spaces; the inclusion is still not full.)

Every Cauchy space is a convergence space; $F \to x$ if the intersection of $F$ with the principal ultrafilter at $x$ is Cauchy. Note that any convergent proper filter must be Cauchy. Conversely, if every Cauchy filter is convergent, then the Cauchy space is called complete.

A Cauchy space $S$ is totally bounded (or precompact) if every proper filter is contained in a Cauchy filter. Equivalently (assuming the Boolean prime ideal theorem), $S$ is totally bounded iff every ultrafilter is Cauchy.

A Cauchy space is compact (as a convergence space) if and only if it is both complete and totally bounded.

References

• Wikipedia.
• Johnstone, Peter T. (1982). Stone Spaces. Cambridge University Press. ISBN 0-521-23893-5.
• Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989.