Finite intersection property

The term "finite intersection property" refers to a property of a family or collection of subsets of a set X, where every finite subcollection of the family has a nonempty intersection. Alternatively, a set A is said to have the finite intersection property if, whenever a collection of closed sets has an empty intersection with A, there exists a finite subcollection of the sets whose intersection with A is also empty.From: Principles of Analysis [2018], General theorems of the Knaster-Kuratowski- Mazurkiewicz type and applications to the existence study in optimization [2020], Advanced Calculus [2019]

Functions of Several Variables

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Published in John Srdjan Petrovic, Advanced Calculus, 2020

John Srdjan Petrovic

10.7.13. We say that a set A has the finite intersection property if, whenever F is a collection of closed sets such that ∩{F: F ∈ F} ∩ A = ø then there exists a finite set F1, F2, …,Fm ∈ F such that ∩i=1mFi ∩ A = ø. Prove that a set A ⊂ ℝn I has the finite intersection property if and only if A is compact.

Miscellaneous Topics

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Published in Hugo D. Junghenn, Principles of Analysis, 2018

Hugo D. Junghenn

By hypothesis, the collection {FT:T∈T} $ \{F_T: T \in \mathcal{T}\} $ has the finite intersection property, hence, by compactness, has a nonempty intersection.

General theorems of the Knaster-Kuratowski- Mazurkiewicz type and applications to the existence study in optimization

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Published in Optimization, 2020

Phan Quoc Khanh, Vo Si Trong Long

‘If’. First, we show that the family has the finite intersection property, i.e. all the finite intersections of sets of this family are nonempty. As F is a g-KKM map, for each , there exists such that for each , Hence, Let , where is the set of vertices of , be defined by . Inclusion (1) shows that is a KKM map. By the KKM principle, Hence, Since is arbitrary, the family has the finite intersection property, and so does the family (with B given in (iii)). As this is a family of compact sets, the total intersection must be nonempty. By the intersectional closedness of F, .

Finite intersection property

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Functions of Several Variables

Miscellaneous Topics

General theorems of the Knaster-Kuratowski- Mazurkiewicz type and applications to the existence study in optimization