nLab category of filters

Definition

A filter $\mathfrak{F}$ on a set $X$ is a set of of subsets of X which has the following properties:

($F_I$) Every subset of X which contains a set of $\mathfrak{F}$ belongs to $\mathfrak{F}$.

($F_{II}$) Every finite intersection of sets of $\mathfrak{F}$ belongs to $\mathfrak{F}$.

Subsets in $\mathfrak{F}$ are called neighbourhoods or $\mathfrak{F}$-big.

A mapping of underlying sets of filters is continuous iff the preimage of a neighbourhood is necessarily a neighbourhood. Note that $\mathfrak{F}\cup \{\emptyset\}$ is a topology on $X$, but this notion of continuity is stronger: it means continuous with dense image.

Let ዋ denote the category of filters and the continuous mappings of underlying sets.

Idea

It is convenient to talk about a filter on a set as a filter structure on a set and think of it as a space with a notion of smallness, in analogy to a topological structure on a set and thinking of it as a (topological) space with a notion of nearness of points.

A filter $\mathfrak{F}$ on a set $X$ gives a precise meaning to phrases “property $P(x)$ holds for all points small enough’‘ and “holds for almost all elements of $X$’’, namely that
$\{ x : P(x) \} \in \mathfrak{F}$, in the same way that the topology (topological structure) on a set gives a precise meaning to the phrase “property $P(x)$ holds for all points sufficiently near a given point $x_0$‘’, namely $\{ x : P(x) \}$ is a neighbourhood of $x_0$.

Thus, for example, in the neighbourhood filter of a point $x_0$ in a topological space, a point $x$ is small iff $x$ is near $x_0$: we think of $x$ near $x_0$ as an approximation to $x_0$ up to a small error.

A number of elementary notions can be formulated using the category of simplicial objects of the category of filters.

Application to analysis and topology

In a topological space $X$, a filter $\mathfrak{F}$ on $X$ converges to a point $x$ of $X$ if every neighbourhood of $x$ belongs to $\mathfrak{F}$, i.e. the map from $\mathfrak{F}$ to the neighbourhood filter of $x$ is continuous. The filter on the empty set is the initial object; the proper filter on a single point is the terminal object.

The concepts of continuous function and uniformly continuous function may be defined quite nicely in terms of the continuity of maps of filters. A map of topological spaces is continuous iff it induces continuous maps of neighbourhood filters. A map of uniform spaces is continuous iff it induces a continuous map of the uniform structures (filters on $M\times M$).

In a metric space $M$, a filter $\mathfrak{F}$ on $M$ is Cauchy if it has elements of arbitrarily small diameter, i.e. the map $\mathfrak{F}\times \mathfrak{F}\to S\times S$ is continuous with respect to the uniformity (filter) on $S\times S$. Then a sequence is a Cauchy sequence iff its eventuality filter is Cauchy, i.e. it induces a continuous map of certain filters. This gives a diagram chasing meaning to the concept of completion of a metric space.

 See also

• filter