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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.
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.
A simple corollary of the theorem is that the Cantor set is nonempty, since it is defined as the intersection of a decreasing nested sequence of sets, each of which is defined as the union of a finite number of closed intervals; hence each of these sets is non-empty, closed, and bounded. In fact, the Cantor set contains uncountably many points.
A finite family of subsets of a finite set is also called a hypergraph. The subject of ... Finite Intersection Property: Additionally, a semiring is a ...
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.
finite intersection property FIP The finite intersection property, abbreviated FIP, says that the intersection of any finite number of elements of a set is non-empty first 1. A set of first category is the same as a meager set: one that is the union of a countable number of nowhere-dense sets. 2. An ordinal of the first class is a finite ordinal 3.
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 ...
A finite union of compact sets is compact. A continuous image of a compact space is compact. [19] The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed); If X is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example). [a]