Finite intersection property

In general topology, a branch of mathematics, a family ${\displaystyle A}$ of subsets of a set ${\displaystyle X}$ is said to have the finite intersection property (FIP) if any finite subfamily of ${\displaystyle {\mathcal {A}}}$ 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.^[1]

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.

Let ${\displaystyle X}$ be a set and ${\displaystyle {\mathcal {A}}}$ a family of subsets of ${\displaystyle X}$ (a subset of the power set of ${\displaystyle X}$). Then ${\displaystyle {\mathcal {A}}}$ is said to have the finite intersection property if the intersection of a finite number of subsets from ${\displaystyle {\mathcal {A}}}$ is always non-empty; it is said to have the strong finite intersection property if that intersection is always infinite.^[1]

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.^[2]

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

If ${\displaystyle {\mathcal {A}}}$ has a non-empty kernel, then it has the finite intersection property trivially. The converse is false in general (although it holds trivially when ${\displaystyle {\mathcal {A}}}$ 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 ${\displaystyle \{\{1,2\},\{2,3\},\{1,3\}\}}$ has non-empty pairwise intersections, but does not possess the finite intersection property. More generally, let ${\displaystyle n}$ be a natural number, let ${\displaystyle X}$ be a set with ${\displaystyle n}$ elements and let ${\displaystyle {\mathcal {A}}}$ consists of those subsets of ${\displaystyle X}$ which contain all elements but one. Then the intersection of fewer than ${\displaystyle n}$ subsets from ${\displaystyle {\mathcal {A}}}$ has non-empty intersection, but ${\displaystyle {\mathcal {A}}}$ lacks the finite intersection property.

If ${\displaystyle A_{0}\supseteq A_{1}\supseteq A_{2}\cdots }$ is a decreasing sequence of non-empty sets, then the family ${\textstyle {\mathcal {A}}=\left\{A_{0},A_{1},A_{2},\ldots \right\}}$ has the finite intersection property (and is even a π–system). If each ${\displaystyle A_{i}}$ is infinite, then ${\textstyle {\mathcal {A}}}$ 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 ${\displaystyle [a,+\infty )}$ for ${\displaystyle a\in \mathbb {R} }$.

The family of all Borel subsets of ${\displaystyle [0,1]}$ with Lebesgue measure 1 has the finite intersection property, as does the family of comeagre sets.^[3][4]

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

If ${\displaystyle K\subseteq X}$ is a non-empty set, then the family ${\displaystyle {\mathcal {A}}=\{S\subseteq X:K\subseteq S\}}$ has the FIP; this family is called the principal filter on ${\textstyle X}$ generated by ${\textstyle K}$. The subset ${\displaystyle {\mathcal {B}}=\{I\subseteq \mathbb {R} :K\subseteq I{\text{ and }}I{\text{ an open interval}}\}}$ has the FIP for much the same reason: the kernels contain the non-empty set ${\textstyle K}$. If ${\textstyle K}$ is an open interval, then the set ${\textstyle K}$ is in fact equal to the kernels of ${\textstyle {\mathcal {A}}}$ or ${\textstyle {\mathcal {B}}}$, 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).

A π–system is a family of sets that is closed under finite intersections of one or more of its sets. For a family of sets ⁠${\displaystyle {\mathcal {A}}}$⁠, the family of sets $${\displaystyle \pi ({\mathcal {A}})=\left\{A_{1}\cap \cdots \cap A_{n}:1\leq n<\infty {\text{ and }}A_{1},\ldots ,A_{n}\in {\mathcal {A}}\right\},}$$which is all finite intersections of one or more sets from ${\displaystyle {\mathcal {A}}}$, is called the π–system generated by ${\textstyle {\mathcal {A}}}$, because it is the smallest π–system having ${\textstyle {\mathcal {A}}}$ as a subset.

The upward closure of ${\displaystyle \pi ({\mathcal {A}})}$ in ${\textstyle X}$ is the set $${\displaystyle \pi ({\mathcal {A}})^{\uparrow X}=\left\{S\subseteq X:P\subseteq S{\text{ for some }}P\in \pi ({\mathcal {A}})\right\}{\text{.}}}$$For any family ${\textstyle {\mathcal {A}}}$, the finite intersection property is equivalent to any of the following:

• The π–system generated by ${\displaystyle {\mathcal {A}}}$ does not have the empty set as an element; that is, ${\displaystyle \varnothing \notin \pi ({\mathcal {A}}).}$
• The set ${\displaystyle \pi ({\mathcal {A}})}$ has the finite intersection property.
• The set ${\displaystyle \pi ({\mathcal {A}})}$ is a (proper)^[5] prefilter.
• The family ${\displaystyle {\mathcal {A}}}$ is a subset of some (proper) prefilter.^[1]
• The upward closure ${\displaystyle \pi ({\mathcal {A}})^{\uparrow X}}$ is a (proper) filter on ${\displaystyle X}$. In this case, ${\displaystyle \pi ({\mathcal {A}})^{\uparrow X}}$ is called the filter on ${\displaystyle X}$ generated by ${\displaystyle {\mathcal {A}}}$, because it is the minimal (with respect to ${\displaystyle \,\subseteq \,}$) filter on ${\displaystyle X}$ that contains ${\displaystyle {\mathcal {A}}}$ as a subset.
• ${\displaystyle {\mathcal {A}}}$ is a subset of some (proper)^[5] filter.^[1]

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.

Another common application is to prove that the real numbers are 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.

Let ${\displaystyle X}$ be non-empty, ${\displaystyle F\subseteq 2^{X}.}$ ${\displaystyle F}$ having the finite intersection property. Then there exists an ${\displaystyle U}$ ultrafilter (in ${\displaystyle 2^{X}}$) such that ${\displaystyle F\subseteq U.}$ This result is known as the ultrafilter lemma.^[8]

• Filter on a set – Family of subsets representing "large" sets
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results
• Neighbourhood system – Concept in mathematics
• Ultrafilter on a set – Maximal proper filter

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• Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Comfort, William Wistar; Negrepontis, Stylianos (1974). The Theory of Ultrafilters. Vol. 211. Berlin Heidelberg New York: Springer-Verlag. ISBN 978-0-387-06604-2. OCLC 1205452.
• Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
• Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750.
• Koutras, Costas D.; Moyzes, Christos; Nomikos, Christos; Tsaprounis, Konstantinos; Zikos, Yorgos (20 October 2021). "On Weak Filters and Ultrafilters: Set Theory From (and for) Knowledge Representation". Logic Journal of the IGPL. 31: 68–95. doi:10.1093/jigpal/jzab030.
• MacIver R., David (1 July 2004). "Filters in Analysis and Topology" (PDF). Archived from the original (PDF) on 2007-10-09. (Provides an introductory review of filters in topology and in metric spaces.)
• Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. (accessible to patrons with print disabilities)
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• "Finite intersection property". PlanetMath.

Families ${\displaystyle {\mathcal {F}}}$ of sets over ${\displaystyle \Omega }$ v t e
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F}}}$	${\displaystyle \varnothing \in {\mathcal {F}}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if ${\displaystyle A_{i}\searrow }$	only if ${\displaystyle A_{i}\nearrow }$
𝜆-system (Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if ${\displaystyle A_{i}\nearrow }$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Filter
Proper filter				Never	Never				Never
Prefilter (Filter base)
Filter subbase
Open Topology							(even arbitrary ${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary ${\displaystyle \cap }$)				Never
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement ${\displaystyle B\setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ A semialgebra is a semiring where every complement ${\displaystyle \Omega \setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F}}}$ and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$