Finite intersection property
In general topology, a branch of mathematics, a family of subsets of a set is said to have the finite intersection property if any finite subfamily of has non-empty intersection. It has the strong finite intersection property 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
Let be a set and a family of subsets of . Then is said to have the finite intersection property if the intersection of a finite number of subsets from 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, from much the same etymology as the sunflower. 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.If has a non-empty kernel, then it has the finite intersection property trivially. The converse is false in general. 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 than requiring pairwise intersection to be non-empty, e.g., the family has non-empty pairwise intersections, but does not possess the finite intersection property. More generally, let be a natural number, let be a set with elements and let consists of those subsets of which contain all elements but one. Then the intersection of fewer than subsets from has non-empty intersection, but lacks the finite intersection property.
End-type constructions
If is a decreasing sequence of non-empty sets, then the family has the finite intersection property. If each is infinite, then admits the strong finite intersection property as well.More generally, any family of non-empty sets which is totally ordered by inclusion has the finite intersection property, and any family of infinite sets which is totally ordered by inclusion has the strong finite intersection property. At the same time, the kernel may be empty: consider the family of subsets for.
"Generic" sets and properties
The family of all Borel subsets of with Lebesgue measure 1 has the finite intersection property, as does the family of comeagre sets.If and, for each positive integer, the subset is precisely all elements of having digit in the th decimal place, then any finite intersection of is non-empty — just take in those finitely many places and in the rest. But the intersection of for all is empty, since no element of has all zero digits.
Generated filters and topologies
If is a non-empty set, then the family has the FIP; this family is called the principal filter on generated by The subset has the FIP for much the same reason: the kernels contain the non-empty set If is an open interval, then the set is in fact equal to the kernels of 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.
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, the family of sets which is all finite intersections of one or more sets from, is called the –system generated by because it is the smallest –system having as a subset.The upward closure of in is the set For any family the finite intersection property is equivalent to any of the following:
- The –system generated by does not have the empty set as an element; that is,
- The set has the finite intersection property.
- The set is a prefilter.
- The family is a subset of some prefilter.
- The upward closure is a filter on In this case, is called the filter on generated by because it is the minimal filter on that contains as a subset.
- is a subset of some 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:- 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.
- The theorem can fail without the compactness condition, as the set of rational numbers shows.
- The theorem can fail without the perfect condition, as any finite space with the discrete topology shows.