the finite intersection property (FIP) if the intersection over any finite subcollection of A {\displaystyle A} is non-empty. It has the strong finite intersection...

non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along with...

downward direction and upward closure reduce to: Closure under finite intersections If A, B ∈ F, then so too is A ∩ B ∈ F. Isotony If A ∈ F and A ⊆ B...

elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition. The main use of σ-algebras...

{B}}}B} is a finite intersection and the filter subbase B {\displaystyle {\mathcal {B}}} has the finite intersection property. A finite prefilter is necessarily...

that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception...

{p}}\supseteq p} and the finite intersection property of G {\displaystyle G} , the set C {\displaystyle C} also has the finite intersection property. Elements of...

based on a property of its members Combinatorial design – Symmetric arrangement of finite sets δ-ring – Ring closed under countable intersections Field of...

Any collection of closed subsets of X with the finite intersection property has nonempty intersection. Every net on X has a convergent subnet (see the...

subsets of the natural numbers that has the strong finite intersection property but has no pseudo-intersection. Kunen, Kenneth (2011), Set theory, Studies in...

E_{n}\in \Sigma \right\}.} A non-empty family of sets has the finite intersection property if and only if the π-system it generates does not contain the...

upwards-closed finite intersection property FIP The finite intersection property, abbreviated FIP, says that the intersection of any finite number of elements...

subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that every k-fold intersection is non-empty has non-empty...

subbase is a non-empty family of sets that has the finite intersection property (i.e. all finite intersections are non-empty). Equivalently, a filter subbase...

the finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and...

extreme points of a set. FFT – fast Fourier transform. FIP – finite intersection property. FOC – first order condition. FOL – first-order logic. fr – boundary...

Relatively compact subspace Heine–Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space...

Albert G. Howson showed that the intersection of two finitely generated subgroups of a free group is again finitely generated. Furthermore, if m {\displaystyle...

uniform properties that are not topological properties. Separated. A uniform space X is separated if the intersection of all entourages is equal to the diagonal...

operations of taking complements in X , {\displaystyle X,} finite unions, and finite intersections. Fields of sets should not be confused with fields in ring...

under finite intersections) is a 𝜎-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply...

finite sets as compact discrete spaces, and then applying the finite intersection property characterization of compactness. In the category of topological...

compact space any collection of closed sets with the finite intersection property has nonempty intersection. The result for locally compact Hausdorff spaces...

if every family of closed subsets having the finite intersection property (FIP) has non-empty intersection (that is, its kernel is not empty). The definition...

closed under both unions and intersections. On the real line R, the family of sets consisting of the empty set and all finite unions of half-open intervals...

there exist δ-rings that are not 𝜎-rings. If the first property is weakened to closure under finite union (that is, A ∪ B ∈ R {\displaystyle A\cup B\in {\mathcal...

the finite sets as compact discrete spaces, and then using the finite intersection property characterization of compactness. Aronszajn tree, for the possible...

this fact, Tychonoff's theorem can be applied; we then use the finite intersection property (FIP) definition of compactness. The proof itself (due to J....

0 {\displaystyle \mu _{0}(\varnothing )=0} and, for every countable (or finite) sequence A 1 , A 2 , … ∈ R {\displaystyle A_{1},A_{2},\ldots \in R} of...

countable. See Cantor's first uncountability proof, and also Finite intersection property#Applications for a topological proof. Manetti, Marco (19 June...