Finite intersection property

In general topology, a branch of mathematics, a family <math>\mathcal{A}</math> of subsets of a set <math>X</math> is said to have the finite intersection property (FIP) if any finite subfamily of <math>\mathcal{A}</math> 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

Let <math>X</math> be a set and <math>\mathcal{A}</math> a family of subsets of <math>X</math> (a subset of the power set of <math>X</math>). Then <math>\mathcal{A}</math> is said to have the finite intersection property if the intersection of a finite number of subsets from <math>\mathcal{A}</math> 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 <math>\mathcal{A}</math> has a non-empty kernel, then it has the finite intersection property trivially. The converse is false in general (although it holds trivially when <math>\mathcal{A}</math> 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 than requiring pairwise intersection to be non-empty, e.g., the family <math>\{\{1,2\}, \{2,3\}, \{1,3\}\}</math> has non-empty pairwise intersections, but does not possess the finite intersection property. More generally, let <math>n</math> be a natural number, let <math>X</math> be a set with <math>n</math> elements and let <math>\mathcal{A}</math> consists of those subsets of <math>X</math> which contain all elements but one. Then the intersection of fewer than <math>n</math> subsets from <math>\mathcal{A}</math> has non-empty intersection, but <math>\mathcal{A}</math> lacks the finite intersection property.

End-type constructions

If <math>A_0 \supseteq A_1 \supseteq A_2 \cdots</math> is a decreasing sequence of non-empty sets, then the family <math display="inline">\mathcal{A} = \left\{A_0, A_1, A_2, \ldots\right\}</math> has the finite intersection property (and is even a –system). If each <math>A_i</math> is infinite, then <math display="inline">\mathcal{A}</math> 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 <math>[a, +\infty)</math> for <math>a \in \R</math>.

"Generic" sets and properties

The family of all Borel subsets of <math>[0, 1]</math> with Lebesgue measure 1 has the finite intersection property, as does the family of comeagre sets.

If <math>X = (0, 1)</math> and, for each positive integer <math>i</math>, the subset <math>X_i</math> is precisely all elements of <math>X</math> having digit <math>0</math> in the <math>i</math><sup>th</sup> decimal place, then any finite intersection of <math>X_i</math> is non-empty — just take <math>0</math> in those finitely many places and <math>1</math> in the rest. But the intersection of <math>X_i</math> for all <math>i \geq 1</math> is empty, since no element of <math>(0, 1)</math> has all zero digits.

Generated filters and topologies

If <math>K \subseteq X</math> is a non-empty set, then the family <math>\mathcal{A}=\{S \subseteq X : K \subseteq S\}</math> has the FIP; this family is called the principal filter on <math display="inline">X</math> generated by The subset <math>\mathcal{B} = \{I \subseteq \R : K \subseteq I \text{ and } I \text{ an open interval}\}</math> has the FIP for much the same reason: the kernels contain the non-empty set If <math display="inline">K</math> is an open interval, then the set <math display="inline">K</math> is in fact equal to the kernels of <math display="inline">\mathcal{A}</math> 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 , the family of sets <math display="block">\pi(\mathcal{A}) = \left\{A_1 \cap \cdots \cap A_n : 1 \leq n < \infty \text{ and } A_1, \ldots, A_n \in \mathcal{A}\right\}, </math>which is all finite intersections of one or more sets from <math>\mathcal{A}</math>, is called the –system generated by because it is the smallest –system having <math display="inline">\mathcal{A}</math> as a subset.

The upward closure of <math>\pi(\mathcal{A})</math> in <math display="inline">X</math> is the set <math display="block">\pi(\mathcal{A})^{\uparrow X} = \left\{S \subseteq X : P \subseteq S \text{ for some } P \in \pi(\mathcal{A})\right\}\text{.}</math>For any family the finite intersection property is equivalent to any of the following:

The –system generated by <math>\mathcal{A}</math> does not have the empty set as an element; that is, <math>\varnothing \notin \pi(\mathcal{A}).</math>
The set <math>\pi(\mathcal{A})</math> has the finite intersection property.
The set <math>\pi(\mathcal{A})</math> is a (proper) prefilter.
The family <math>\mathcal{A}</math> is a subset of some (proper) prefilter.
The upward closure <math>\pi(\mathcal{A})^{\uparrow X}</math> is a (proper) filter on In this case, <math>\pi(\mathcal{A})^{\uparrow X}</math> is called the filter on <math>X</math> generated by because it is the minimal (with respect to <math>\,\subseteq\,</math>) filter on <math>X</math> that contains <math>\mathcal{A}</math> as a subset.
<math>\mathcal{A}</math> is a subset of some (proper)

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.

Ultrafilters

Let <math>X</math> be non-empty, <math>F \subseteq 2^X.</math> <math>F</math> having the finite intersection property. Then there exists an <math>U</math> ultrafilter (in <math>2^X</math>) such that <math>F \subseteq U.</math> This result is known as the ultrafilter lemma.

References

Notes

Citations

General sources

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