{"history":[],"article":{"id":"5d3faa1d-3c5f-4f7e-9296-89bac96a83fe","title":"Finite intersection property","slug":"finite-intersection-property","content":":::summary\n:::\n\n# \nIn [general topology](https://bliptext.com/articles/general-topology), a branch of [mathematics](https://bliptext.com/articles/mathematics), a [non-empty](https://bliptext.com/articles/empty-set) family A of [subsets](https://bliptext.com/articles/sub[set](https://bliptext.com/articles/set-mathematics)) of a [set](https://bliptext.com/articles/set-mathematics) X is said to have the **finite [intersection](https://bliptext.com/articles/intersection-set-theory) property** (FIP) if the [intersection](https://bliptext.com/articles/intersection-set-theory) over any finite subcollection of A is [non-empty](https://bliptext.com/articles/empty-set).\nIt has the **strong finite intersection property** (SFIP) if the intersection over any finite subcollection of A is infinite.\nSets with the finite intersection property are also called **centered systems** and **filter subbases**.\nThe finite intersection property can be used to reformulate topological [compactness](https://bliptext.com/articles/compactness) in terms of [closed sets](https://bliptext.com/articles/closed-set); this is its most prominent application.\nOther applications include proving that certain [perfect sets](https://bliptext.com/articles/perfect-set) are uncountable, and the construction of [ultrafilters](https://bliptext.com/articles/ultrafilter-set-theory).\n\n# Definition\nLet X be a set and \\mathcal{A} a [nonempty](https://bliptext.com/articles/nonempty) [family of subsets](https://bliptext.com/articles/family-of-sets) of X; that is, \\mathcal{A} is a [subset](https://bliptext.com/articles/subset) of the [power set](https://bliptext.com/articles/power-set) of X. Then \\mathcal{A} is said to have the finite intersection property if every [nonempty](https://bliptext.com/articles/nonempty) finite subfamily has [nonempty](https://bliptext.com/articles/nonempty) intersection; it is said to have the strong finite intersection property if that intersection is always infinite.\nIn symbols, \\mathcal{A} has the FIP if, for any choice of a finite nonempty subset \\mathcal{B} of \\mathcal{A}, there must exist a point Likewise, \\mathcal{A} has the SFIP if, for every choice of such \\mathcal{B}, there are infinitely many such x.\nIn the study of [filters](https://bliptext.com/articles/filter-set-theory), the common intersection of a family of sets is called a [kernel](https://bliptext.com/articles/kernel-filters), from much the same etymology as the [sunflower](https://bliptext.com/articles/sunflower-mathematics).\nFamilies with empty kernel are called [free](https://bliptext.com/articles/free-filter); those with nonempty kernel, [fixed](https://bliptext.com/articles/fixed-filter-math).\n\n# Families of examples and non-examples\nThe empty set cannot belong to any collection with the finite intersection property.\nA sufficient condition for the FIP intersection property is a nonempty kernel.\nThe converse is generally false, but holds for finite families; that is, if \\mathcal{A} is finite, then \\mathcal{A} has the finite intersection property if and only if it is fixed.\n\n## Pairwise intersection\nThe finite intersection property is strictly stronger than pairwise intersection; the family has pairwise intersections, but not the FIP.\nMore generally, let be a positive integer greater than unity,, and.\nThen any subset of \\mathcal{A} with fewer than n elements has nonempty intersection, but \\mathcal{A} lacks the FIP.\n\n## End-type constructions\nIf is a decreasing sequence of non-empty sets, then the family has the finite intersection property (and is even a [π–system](https://bliptext.com/articles/pi-system)).\nIf the inclusions are [strict](https://bliptext.com/articles/strict-subset), then \\mathcal{A} admits the strong finite intersection property as well.\nMore generally, any \\mathcal{A} that is [totally ordered](https://bliptext.com/articles/total-order) by inclusion has the FIP.\nAt the same time, the [kernel](https://bliptext.com/articles/) of \\mathcal{A} may be empty: if, then the [kernel](https://bliptext.com/articles/) of \\mathcal{A} is the [empty set](https://bliptext.com/articles/empty-set).\nSimilarly, the family of intervals also has the (S)FIP, but empty kernel.\n\n## \"Generic\" sets and properties\nThe family of all [Borel subsets](https://bliptext.com/articles/borel-set) of [0, 1] with [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) 1 has the FIP, as does the family of [comeagre](https://bliptext.com/articles/comeagre) sets.\nIf X is an infinite set, then the [Fréchet filter](https://bliptext.com/articles/fr-chet-filter) (the family ) has the FIP.\nAll of these are [free filters](https://bliptext.com/articles/free-filter); they are upwards-closed and have empty infinitary intersection.\nIf X = (0, 1) and, for each positive integer i, the subset X_i is precisely all elements of X having [digit](https://bliptext.com/articles/digit-math) 0 in the ith [decimal place](https://bliptext.com/articles/decimal-place), then any finite intersection of X_i is non-empty — just take 0 in those finitely many places and 1 in the rest.\nBut the intersection of X_i for all i \\geq 1 is empty, since no element of (0, 1) has all zero digits.\n\n## Extension of the ground set\nThe (strong) finite intersection property is a characteristic of the family \\mathcal{A}, not the ground set X. If a family \\mathcal{A} on the set X admits the (S)FIP and, then \\mathcal{A} is also a family on the set Y with the FIP (resp. SFIP).\n\n## Generated filters and topologies\nIf are sets with then the family has the FIP; this family is called the principal filter on X generated by K. The subset has the FIP for much the same reason: the kernels contain the non-empty set K. If K is an open interval, then the set K is in fact equal to the kernels of \\mathcal{A} or \\mathcal{B}, and so is an element of each filter.\nBut in general a filter's kernel need not be an element of the filter.\nA [proper filter on a set](https://bliptext.com/articles/proper-filter-set-theory) has the finite intersection property.\nEvery [neighbourhood subbasis](https://bliptext.com/articles/neighbourhood-subbasis) at a point in a [topological space](https://bliptext.com/articles/topological-space) has the FIP, and the same is true of every [neighbourhood basis](https://bliptext.com/articles/neighbourhood-basis) and every [neighbourhood filter](https://bliptext.com/articles/neighbourhood-filter) at a point (because each is, in particular, also a [neighbourhood subbasis](https://bliptext.com/articles/neighbourhood-subbasis)).\n\n# Relationship to π-systems and filters\nA [π–system](https://bliptext.com/articles/pi-system) is a non-empty family of sets that is closed under finite intersections.\nThe set of all finite intersections of one or more sets from \\mathcal{A} is called the [π–system](https://bliptext.com/articles/pi-system) generated by \\mathcal{A}, because it is the [smallest](https://bliptext.com/articles/minimal-element) [π–system](https://bliptext.com/articles/pi-system) having \\mathcal{A} as a subset.\nThe upward closure of in X is the set\nFor any family \\mathcal{A}, the finite intersection property is equivalent to any of the following:\n

The [π–system](https://bliptext.com/articles/pi-system) generated by \\mathcal{A} does not have the [empty set](https://bliptext.com/articles/empty-set) as an element; that is,\n
The set has the finite intersection property.
The set is a (proper) [prefilter](https://bliptext.com/articles/prefilter).
The family \\mathcal{A} is a subset of some (proper) [prefilter](https://bliptext.com/articles/prefilter).
The upward closure is a [(proper) filter](https://bliptext.com/articles/proper-filter-set-theory) on X. In this case, is called the filter on X generated by \\mathcal{A}, because it is the minimal (with respect to ) filter on X that contains \\mathcal{A} as a subset.
\\mathcal{A} is a subset of some (proper) filter.

\n\n# Applications\n\n## Compactness\nThe finite intersection property is useful in formulating an alternative definition of [compactness](https://bliptext.com/articles/compact-space):\nThis formulation of compactness is used in some proofs of [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem).\n\n## Uncountability of perfect spaces\nAnother common application is to prove that the [real numbers](https://bliptext.com/articles/real-number) are [uncountable](https://bliptext.com/articles/uncountable-set).\nAll the conditions in the statement of the theorem are necessary:\n\n## Ultrafilters\nLet X be non-empty, F having the finite intersection property.\nThen there exists an U [ultrafilter](https://bliptext.com/articles/ultrafilter-set-theory) (in 2^X) such that This result is known as the [ultrafilter](https://bliptext.com/articles/ultrafilter-set-theory) lemma.\n\n## Citations\n\n## General sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:21:00.930Z","updated_at":"2025-01-25T19:21:00.930Z","search_vector":"'finit':1 'intersect':2 'properti':3","search_content":" \n \n\n# \nIn [general topology](https://bliptext.com/articles/general-topology), a branch of [mathematics](https://bliptext.com/articles/mathematics), a [non-empty](https://bliptext.com/articles/empty-set) family A of [sub[set](https://bliptext.com/articles/set-mathematics)s](https://bliptext.com/articles/sub[set](https://bliptext.com/articles/set-mathematics)) of a [set](https://bliptext.com/articles/set-mathematics) X is said to have the **finite [intersection](https://bliptext.com/articles/intersection-set-theory) property** (FIP) if the [intersection](https://bliptext.com/articles/intersection-set-theory) over any finite subcollection of A is [non-empty](https://bliptext.com/articles/empty-set).\nIt has the **strong finite intersection property** (SFIP) if the intersection over any finite subcollection of A is infinite.\nSets with the finite intersection property are also called **centered systems** and **filter subbases**.\nThe finite intersection property can be used to reformulate topological [compactness](https://bliptext.com/articles/compactness) in terms of [closed sets](https://bliptext.com/articles/closed-set); this is its most prominent application.\nOther applications include proving that certain [perfect sets](https://bliptext.com/articles/perfect-set) are uncountable, and the construction of [ultrafilters](https://bliptext.com/articles/ultrafilter-set-theory).\n\n# Definition\nLet X be a set and \\mathcal{A} a [nonempty](https://bliptext.com/articles/nonempty) [family of [subset](https://bliptext.com/articles/subset)s](https://bliptext.com/articles/family-of-sets) of X; that is, \\mathcal{A} is a [subset](https://bliptext.com/articles/subset) of the [power set](https://bliptext.com/articles/power-set) of X. Then \\mathcal{A} is said to have the finite intersection property if every [nonempty](https://bliptext.com/articles/nonempty) finite subfamily has [nonempty](https://bliptext.com/articles/nonempty) intersection; it is said to have the strong finite intersection property if that intersection is always infinite.\nIn symbols, \\mathcal{A} has the FIP if, for any choice of a finite nonempty subset \\mathcal{B} of \\mathcal{A}, there must exist a point Likewise, \\mathcal{A} has the SFIP if, for every choice of such \\mathcal{B}, there are infinitely many such x.\nIn the study of [filters](https://bliptext.com/articles/filter-set-theory), the common intersection of a family of sets is called a [kernel](https://bliptext.com/articles/kernel-filters), from much the same etymology as the [sunflower](https://bliptext.com/articles/sunflower-mathematics).\nFamilies with empty kernel are called [free](https://bliptext.com/articles/free-filter); those with nonempty kernel, [fixed](https://bliptext.com/articles/fixed-filter-math).\n\n# Families of examples and non-examples\nThe empty set cannot belong to any collection with the finite intersection property.\nA sufficient condition for the FIP intersection property is a nonempty kernel.\nThe converse is generally false, but holds for finite families; that is, if \\mathcal{A} is finite, then \\mathcal{A} has the finite intersection property if and only if it is fixed.\n\n## Pairwise intersection\nThe finite intersection property is strictly stronger than pairwise intersection; the family has pairwise intersections, but not the FIP.\nMore generally, let be a positive integer greater than unity,, and.\nThen any subset of \\mathcal{A} with fewer than n elements has nonempty intersection, but \\mathcal{A} lacks the FIP.\n\n## End-type constructions\nIf is a decreasing sequence of non-empty sets, then the family has the finite intersection property (and is even a [π–system](https://bliptext.com/articles/pi-system)).\nIf the inclusions are [strict](https://bliptext.com/articles/strict-subset), then \\mathcal{A} admits the strong finite intersection property as well.\nMore generally, any \\mathcal{A} that is [totally ordered](https://bliptext.com/articles/total-order) by inclusion has the FIP.\nAt the same time, the [kernel](https://bliptext.com/articles/) of \\mathcal{A} may be empty: if, then the [kernel](https://bliptext.com/articles/) of \\mathcal{A} is the [empty set](https://bliptext.com/articles/empty-set).\nSimilarly, the family of intervals also has the (S)FIP, but empty kernel.\n\n## \"Generic\" sets and properties\nThe family of all [Borel subsets](https://bliptext.com/articles/borel-set) of [0, 1] with [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) 1 has the FIP, as does the family of [comeagre](https://bliptext.com/articles/comeagre) sets.\nIf X is an infinite set, then the [Fréchet filter](https://bliptext.com/articles/fr-chet-filter) (the family ) has the FIP.\nAll of these are [free filters](https://bliptext.com/articles/free-filter); they are upwards-closed and have empty infinitary intersection.\nIf X = (0, 1) and, for each positive integer i, the subset X_i is precisely all elements of X having [digit](https://bliptext.com/articles/digit-math) 0 in the ith [decimal place](https://bliptext.com/articles/decimal-place), then any finite intersection of X_i is non-empty — just take 0 in those finitely many places and 1 in the rest.\nBut the intersection of X_i for all i \\geq 1 is empty, since no element of (0, 1) has all zero digits.\n\n## Extension of the ground set\nThe (strong) finite intersection property is a characteristic of the family \\mathcal{A}, not the ground set X. If a family \\mathcal{A} on the set X admits the (S)FIP and, then \\mathcal{A} is also a family on the set Y with the FIP (resp. SFIP).\n\n## Generated filters and topologies\nIf are sets with then the family has the FIP; this family is called the principal filter on X generated by K. The subset has the FIP for much the same reason: the kernels contain the non-empty set K. If K is an open interval, then the set K is in fact equal to the kernels of \\mathcal{A} or \\mathcal{B}, and so is an element of each filter.\nBut in general a filter's kernel need not be an element of the filter.\nA [proper filter on a set](https://bliptext.com/articles/proper-filter-set-theory) has the finite intersection property.\nEvery [neighbourhood subbasis](https://bliptext.com/articles/neighbourhood-subbasis) at a point in a [topological space](https://bliptext.com/articles/topological-space) has the FIP, and the same is true of every [neighbourhood basis](https://bliptext.com/articles/neighbourhood-basis) and every [neighbourhood filter](https://bliptext.com/articles/neighbourhood-filter) at a point (because each is, in particular, also a [neighbourhood subbasis](https://bliptext.com/articles/neighbourhood-subbasis)).\n\n# Relationship to π-systems and filters\nA [π–system](https://bliptext.com/articles/pi-system) is a non-empty family of sets that is closed under finite intersections.\nThe set of all finite intersections of one or more sets from \\mathcal{A} is called the [π–system](https://bliptext.com/articles/pi-system) generated by \\mathcal{A}, because it is the [smallest](https://bliptext.com/articles/minimal-element) [π–system](https://bliptext.com/articles/pi-system) having \\mathcal{A} as a subset.\nThe upward closure of in X is the set\nFor any family \\mathcal{A}, the finite intersection property is equivalent to any of the following:\n The [π–system](https://bliptext.com/articles/pi-system) generated by \\mathcal{A} does not have the [empty set](https://bliptext.com/articles/empty-set) as an element; that is,\n \n The set has the finite intersection property. \n The set is a (proper) [prefilter](https://bliptext.com/articles/prefilter). \n The family \\mathcal{A} is a subset of some (proper) [prefilter](https://bliptext.com/articles/prefilter). \n The upward closure is a [(proper) filter](https://bliptext.com/articles/proper-filter-set-theory) on X. In this case, is called the filter on X generated by \\mathcal{A}, because it is the minimal (with respect to ) filter on X that contains \\mathcal{A} as a subset. \n \\mathcal{A} is a subset of some (proper) filter. \n\n# Applications\n\n## Compactness\nThe finite intersection property is useful in formulating an alternative definition of [compactness](https://bliptext.com/articles/compact-space):\nThis formulation of compactness is used in some proofs of [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem).\n\n## Uncountability of perfect spaces\nAnother common application is to prove that the [real numbers](https://bliptext.com/articles/real-number) are [uncountable](https://bliptext.com/articles/uncountable-set).\nAll the conditions in the statement of the theorem are necessary:\n\n## Ultrafilters\nLet X be non-empty, F having the finite intersection property.\nThen there exists an U [ultrafilter](https://bliptext.com/articles/ultrafilter-set-theory) (in 2^X) such that This result is known as the [ultrafilter](https://bliptext.com/articles/ultrafilter-set-theory) lemma.\n\n## Citations\n\n## General sources","revisionCount":0},"hasMore":false,"nextCursor":null}

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