In general topology, a branch of mathematics of subsets of a set is said to have the finite intersection property has non-empty intersection. It has the strong finite intersection property (SFIP) if any finite subfamily has infinite intersection. Sets with the finite intersection property are also called centered systems and filter subbases.

The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters.

Definition

a family of subsets (a subset of the power set is always non-empty; it is said to have the strong finite intersection property if that intersection is always infinite.

In the study of filters, the intersection of a family of sets is called its kernel. Families with empty kernel are called free; those with nonempty kernel, fixed.

Examples and non-examples

The empty set cannot belong to any family with the finite intersection property.

is finite). For example, the family of all cofinite subsets of a fixed infinite set — the Fréchet filter — has the finite intersection property, although its kernel is empty. More generally, any proper filter has the finite intersection property.

The finite intersection property is strictly stronger lacks the finite intersection property.

End-type constructions

has the finite intersection property (and is even a –system admits the strong finite intersection property as well.

More generally, any family of non-empty sets which is totally ordered.

"Generic" sets and properties

The family of all Borel subsets with Lebesgue measure 1 has the finite intersection property, as does the family of comeagre sets.

having digitth decimal place is non-empty has all zero digits.

Generated filters and topologies

generated by has the FIP for much the same reason: the kernels contain the non-empty set or and so is an element of each filter. But in general a filter's kernel need not be an element of the filter.

A proper filter has the finite intersection property. Every neighbourhood subbasis at a point in a topological space has the FIP, and the same is true of every neighbourhood basis and every neighbourhood filter at a point (because each is, in particular, also a neighbourhood subbasis).

Relationship to -systems and filters

A –system is a family of sets that is closed under finite intersections of one or more of its sets. For a family of sets , is called the –system generated by because it is the smallest as a subset.

For any family the finite intersection property is equivalent to any of the following:

• The –system does not have the empty set

• has the finite intersection property.

• is a (proper) prefilter.

• is a subset of some (proper) prefilter.

• is a (proper) filter on generated by as a subset.

• is a subset of some (proper) filter.

Applications

Compactness

The finite intersection property is useful in formulating an alternative definition of compactness:

This formulation of compactness is used in some proofs of Tychonoff's theorem.

Uncountability of perfect spaces

Another common application is to prove that the real numbers are uncountable. Note that a subset of a topological space is perfect if it is closed and has the property that no one-point subset is open. Examples of failures:

1. The theorem can fail without the Hausdorff condition; a countable set with at least two points and with the indiscrete topology is perfect and compact, but is not uncountable.

2. The theorem can fail without the compactness condition, as the set of rational numbers shows.

3. The theorem can fail without the perfect condition, as any finite space with the discrete topology shows.

Ultrafilters

ultrafilter This result is known as the ultrafilter lemma.

See also

• - Generalization to lattices

References

Notes

Citations

General sources

• (Provides an introductory review of filters in topology and in metric spaces.)

External links