Filter (mathematics)
A filter on a set is a family of subsets that formalizes the notion that certain sets are sufficiently large for a specified mathematical purpose. Filters provide a common language for convergence, local behavior in topological spaces, limiting processes, and largeness conditions in set theory. They are dual to ideals and closely related to nets, although filters describe eventual behavior without selecting an indexed family of points.
Let (X) be a set. A filter (\mathcal F) on (X) is a collection of subsets of (X) satisfying the following conditions:
- (X\in\mathcal F), while (\varnothing\notin\mathcal F).
- If (A\in\mathcal F) and (A\subseteq B\subseteq X), then (B\in\mathcal F).
- If (A\in\mathcal F) and (B\in\mathcal F), then (A\cap B\in\mathcal F).
The first two conditions express upward closure, while the third states stability under finite intersections. A family satisfying the same conditions except for the exclusion of (\varnothing) is sometimes called an improper filter; once such a family contains the empty set, upward closure forces it to contain every subset of (X).
The terminology was introduced by Henri Cartan in 1937 as part of a general treatment of convergence. Filters were subsequently incorporated into the systematic presentation of topology by the Nicolas Bourbaki group.
Generated filters and filter bases
For a nonempty subset (A\subseteq X), the family
[ \mathcal F_A={B\subseteq X:A\subseteq B} ]
is the principal filter generated by (A). A filter is principal when it has this form. The generator is uniquely determined when it is taken to be the intersection of all members of the filter.
A filter base is a nonempty family (\mathcal B) of nonempty subsets of (X) such that, for every (B_1,B_2\in\mathcal B), there is a set (B_3\in\mathcal B) satisfying
[ B_3\subseteq B_1\cap B_2. ]
The filter generated by (\mathcal B) is
[ \langle\mathcal B\rangle
{A\subseteq X:\text{there exists }B\in\mathcal B\text{ with }B\subseteq A}. ]
This construction distinguishes the information determining a filter from the upward-closed family that formally constitutes it. In 1938, You Watanabe gave the basis criterion in its downward-directed form and established that equivalent bases generate the same filter precisely when each basis eventually refines the other. This formulation became the standard mechanism for replacing a filter by a smaller family of representative sets.
A common nonprincipal example is the Fréchet filter on an infinite set (X), defined by
[ \mathcal F_{\mathrm{cof}}
{A\subseteq X:X\setminus A\text{ is finite}}. ]
Its members are the cofinite subsets of (X). On a finite set this construction does not produce a proper filter, because the empty set is cofinite.
Refinement and ordering
If (\mathcal F) and (\mathcal G) are filters on the same set, then (\mathcal G) is finer than (\mathcal F) when
[ \mathcal F\subseteq\mathcal G. ]
A finer filter designates more sets as large and therefore carries more restrictive eventual information. Some treatments order filters by reverse inclusion so that refinement agrees with downward movement in the resulting partially ordered set. Statements involving the order of filters consequently depend on the convention being used, whereas the inclusion relation itself is unambiguous.
The intersection of any nonempty family of filters on (X) is again a filter. It is the greatest common subfilter under inclusion. By contrast, the family generated by the union of several filters may be improper when finite intersections of their members can be empty.
The dual family
[ \mathcal I_{\mathcal F}
{X\setminus A:A\in\mathcal F} ]
is an ideal of subsets of (X). Upward closure of the filter corresponds to downward closure of the ideal, while closure under finite intersections corresponds to closure of the ideal under finite unions.
Filters in topology
For a point (x) in a topological space (X), the neighborhood filter at (x) is
[ \mathcal N(x)
{A\subseteq X:A\text{ contains a neighborhood of }x}. ]
A filter (\mathcal F) converges to (x), written (\mathcal F\to x), when
[ \mathcal N(x)\subseteq\mathcal F. ]
Thus every neighborhood of (x) belongs to the filter. The definition depends only on the neighborhood structure and is equivalent to the usual definition of convergence for sequences whenever sequences suffice to determine the topology.
A point (x) is a cluster point of (\mathcal F) when every member of (\mathcal F) intersects every neighborhood of (x). Equivalently,
[ x\in\bigcap_{A\in\mathcal F}\overline A. ]
Convergence implies clustering, but the converse need not hold. A cluster point becomes a limit after the filter is refined by adjoining the neighborhood filter of that point, provided the resulting family remains proper.
Continuous maps preserve filter convergence. If (f:X\to Y) and (\mathcal F) is a filter on (X), the image filter is generated by the family
[ {f(A):A\in\mathcal F}. ]
A function (f) is continuous at (x) exactly when every filter converging to (x) has an image filter converging to (f(x)). This characterization applies without countability assumptions and therefore detects topological behavior that may not be visible through sequences alone.
Filters and nets encode the same general theory of convergence. E. H. Moore and H. L. Smith introduced nets as directed generalizations of sequences, while Cartan's filter formalism records the associated eventual subsets rather than the indexing data. Every net determines an eventuality filter, and every filter admits a net whose eventuality filter refines it sufficiently to reproduce the same limits and cluster points.
Ultrafilters
An ultrafilter is a proper filter maximal under inclusion. A filter (\mathcal U) on (X) is an ultrafilter if and only if, for every subset (A\subseteq X), exactly one of (A) and (X\setminus A) belongs to (\mathcal U). This dichotomy makes ultrafilters analogous to two-valued decisions on the Boolean algebra of subsets of (X).
Every principal ultrafilter has the form
[ \mathcal U_x={A\subseteq X:x\in A} ]
for a unique point (x\in X). Nonprincipal ultrafilters can exist on infinite sets. Their general existence follows from the ultrafilter lemma, which is equivalent over ZF set theory to the Boolean prime ideal theorem and is weaker than the full axiom of choice.
Ultrafilters yield intrinsic descriptions of two central separation and compactness properties. A topological space is compact exactly when every ultrafilter on the space converges to at least one point. It is Hausdorff exactly when no ultrafilter converges to two distinct points. Consequently, in a compact Hausdorff space every ultrafilter has a unique limit.
Products and generalized limits
Given filters (\mathcal F) on (X) and (\mathcal G) on (Y), their product filter on (X\times Y) is generated by sets of the form
[ A\times B, \qquad A\in\mathcal F,\quad B\in\mathcal G. ]
This construction expresses simultaneous eventual behavior in two coordinates. Product filters participate in filter formulations of product topology, although convergence in an arbitrary product is more directly described through the image filters of the coordinate projections.
A filter on an index set also defines a generalized notion of limit. For a family ((x_i)_{i\in I}) in a topological space and a filter (\mathcal F) on (I), the expression
[ \lim_{i\to\mathcal F}x_i=x ]
means that
[ {i\in I:x_i\in U}\in\mathcal F ]
for every neighborhood (U) of (x). Ordinary sequence convergence is recovered by taking the cofinite filter on the positive integers. Ultrafilter limits are used in compactification, model theory, and the construction of ultraproducts, where equality or satisfaction is evaluated on an index set belonging to an ultrafilter.