Filters in topology

This establishes a relationship between filters and prefilters that may often be exploited to allow one to use whichever of these two notions is more technically convenient.

Consequently, subordination also plays an important role in many concepts that are related to convergence, such as cluster points and limits of functions.

Filters were introduced by Henri Cartan in 1937[1] and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith.

provide a framework that seamlessly ties together fundamental topological concepts such as topological spaces (via neighborhood filters), neighborhood bases, convergence, various limits of functions, continuity, compactness, sequences (via sequential filters), the filter equivalent of "subsequence" (subordination), uniform spaces, and more; concepts that otherwise seem relatively disparate and whose relationships are less clear.

But there are many spaces where sequences can not be used to describe even basic topological properties like closure or continuity.

This failure of sequences was the motivation for defining notions such as nets and filters, which never fail to characterize topological properties.

The properties that these families share led to the notion of a filter base, also called a prefilter, which by definition is any family having the minimal properties necessary and sufficient for it to generate a filter via taking its upward closure.

One example is the universal property of inverse limits, which is defined in terms of composition of functions rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is the Cartesian product).

Special types of filters called ultrafilters have many useful properties that can significantly help in proving results.

One downside of nets is their dependence on the directed sets that constitute their domains, which in general may be entirely unrelated to the space

When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author.

and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality

[7] The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.

Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence.

This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in the construction of completions of uniform spaces via Cauchy filters).

But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.

Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful.

So to gain understanding and intuition about how filters (and prefilter) relate to concepts in topology, the "downward" property is usually the one to concentrate on.

is called primitive[45] if it is the set of limit points of some ultrafilter (or equivalently, some ultra prefilter).

The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under

This section will describe the relationships between prefilters and nets in great detail because of how important these details are applying filters to topology − particularly in switching from utilizing nets to utilizing filters and vice verse.

However, in 1955 Bruns and Schmidt discovered[48] a construction (detailed here: Filter (set theory)#Partially ordered net) that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.

), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered.

[53] Stephen Willard introduced in 1970 his own variant ("Willard-subnet") of Kelley's definition of subnet.

AA-subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.

is a prefilter (or equivalently, a π-system) if and only if it has the finite intersection property (that is, it is a filter subbase), in which case a subset

[45] Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces: Let

For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters.

The set of all minimal Cauchy filters on a Hausdorff topological vector space (TVS)

(with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected.