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Let be a set and a nonempty family of subsets of ; that is, is a nonempty subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.
We prove the finite version, using Radon's theorem as in the proof by Radon (1921).The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only if every finite subcollection has a non-empty intersection (once you fix a single set, the intersection of all others ...
It is also equal to number of hyperplanes in general position which are needed to have an intersection with the algebraic set, which is a finite number of points. The degree of the ideal and of its associated algebraic set is the number of points of this finite intersection, counted with multiplicity.
For the usual base for this topology, every finite intersection of basic open sets is a basic open set. The Zariski topology of is the topology that has the algebraic sets as closed sets. It has a base formed by the set complements of algebraic hypersurfaces.
It uses the facts that (1) in such a space every point has a local base of closed compact neighborhoods; and (2) in a compact space any collection of closed sets with the finite intersection property has nonempty intersection. The result for locally compact Hausdorff spaces is a special case, as such spaces are regular.
If is finite then every ultrafilter is a discrete filter at a point; consequently, free ultrafilters can only exist on infinite sets. In particular, if is finite then the ultrafilter lemma can be proven from the axioms ZF. The existence of free ultrafilter on infinite sets can be proven if the axiom of choice is assumed.
On the real line R, the family of sets consisting of the empty set and all finite unions of half-open intervals of the form (a, b], with a, b ∈ R is a ring in the measure-theoretic sense. If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed under both unions and intersections.
A π-system is a non-empty collection of sets that is closed under non-empty finite intersections, which is equivalent to containing the intersection of any two of its elements. If every set in this π -system is a subset of Ω {\displaystyle \Omega } then it is called a π -system on Ω . {\displaystyle \Omega .}