Finite intersection property

In general topology, a branch of mathematics, a collection A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of A is infinite.

A centered system of sets is a collection of sets with the finite intersection property.

Definition

Let ${\displaystyle X}$ be a set and let ${\displaystyle {\mathcal {A}}=\{A_{i}\}_{i\in I}}$ a family of subsets of ${\displaystyle X}$ indexed by an arbitrary set ${\displaystyle I}$. Then the collection ${\displaystyle {\mathcal {A}}}$ has the finite intersection property (FIP), if any finite subcollection ${\displaystyle J\subseteq I}$ has non-empty intersection, that is ${\displaystyle \bigcap _{i\in J}A_{i}}$ is a non-empty subset of ${\displaystyle X.}$

Discussion

The empty set cannot belong to any collection with the finite intersection property. The condition is trivially satisfied if the intersection over the entire collection is non-empty (in particular, if the collection itself is empty), and it is also trivially satisfied if the collection is nested, meaning that the collection is totally ordered by inclusion (equivalently, for any finite subcollection, a particular element of the subcollection is contained in all the other elements of the subcollection), e.g. the nested sequence of intervals (0, 1/n). However, these are not the only possibilities. For example, if X = (0, 1) and for each positive integer i, X_i is the set of elements of X having a decimal expansion with digit 0 in the ith decimal place, then any finite intersection is non-empty (just take 0 in those finitely many places and 1 in the rest), but the intersection of all X_i for i ≥ 1 is empty, since no element of (0, 1) has all zero digits.

The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets having the finite intersection property has non-empty intersection.[1][2] This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers (see next section).

Applications

Theorem. Let X be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open. Then X is uncountable.

Proof. We will show that if U ⊆ X is non-empty and open, and if x is a point of X, then there is a neighbourhood V ⊂ U whose closure does not contain x (x may or may not be in U). Choose y in U different from x (if x is in U, then there must exist such a y for otherwise U would be an open one point set; if x is not in U, this is possible since U is non-empty). Then by the Hausdorff condition, choose disjoint neighbourhoods W and K of x and y respectively. Then K ∩ U will be a neighbourhood of y contained in U whose closure doesn't contain x as desired.

Now suppose f: N → X is a bijection, and let {x_i : i ∈ N} denote the image of f. Let X be the first open set and choose a neighbourhood U₁ ⊂ X whose closure does not contain x₁. Secondly, choose a neighbourhood U₂ ⊂ U₁ whose closure does not contain x₂. Continue this process whereby choosing a neighbourhood U_n+1 ⊂ U_n whose closure does not contain x_n+1. Then the collection {U_i : i ∈ N} satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of X. Therefore, there is a point x in this intersection. No x_i can belong to this intersection because x_i does not belong to the closure of U_i. This means that x is not equal to x_i for all i and f is not surjective; a contradiction. Therefore, X is uncountable.

All the conditions in the statement of the theorem are necessary:

1. We cannot eliminate the Hausdorff condition; a countable set (with at least two points) with the indiscrete topology is compact, has more than one point, and satisfies the property that no one point sets are open, but is not uncountable.

2. We cannot eliminate the compactness condition, as the set of rational numbers shows.

3. We cannot eliminate the condition that one point sets cannot be open, as any finite space with the discrete topology shows.

Corollary. Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable.

Corollary. Every perfect, locally compact Hausdorff space is uncountable.

Proof. Let X be a perfect, compact, Hausdorff space, then the theorem immediately implies that X is uncountable. If X is a perfect, locally compact Hausdorff space that is not compact, then the one-point compactification of X is a perfect, compact Hausdorff space. Therefore, the one point compactification of X is uncountable. Since removing a point from an uncountable set still leaves an uncountable set, X is uncountable as well.

Examples

A proper filter on a set has the finite intersection property.

Theorems

Let X be non-empty, F ⊆ 2^X, F having the finite intersection property. Then there exists an U ultrafilter (in 2^X) such that F ⊆ U.

See details and proof in Csirmaz & Hajnal (1994).[3] This result is known as the ultrafilter lemma.

Variants

A family of sets A has the strong finite intersection property (SFIP), if every finite subfamily of A has infinite intersection.

References

1. Munkres, James (2004). Topology. New Dehli: Prentice-Hall of India. p. 169. ISBN 978-81-203-2046-8.
2. "A space is compact iff any family of closed sets having fip has non-empty intersection". PlanetMath.
3. Csirmaz, László; Hajnal, András (1994), Matematikai logika (In Hungarian), Budapest: Eötvös Loránd University.

• "Finite intersection property". PlanetMath.