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It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. Sets with the finite intersection property are also called centered systems and filter subbases. [1] The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most ...
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 ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions ...
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 free ultrafilter exists on a set if and only if is infinite. Every proper filter is equal to the intersection of all ultrafilters containing it. [4] Since there are filters that are not ultra, this shows that the intersection of a family of ultrafilters need not be ultra.
The intersection property also allows one to define the closure of a set in a space , which is defined as the smallest closed subset of that is a superset of . Specifically, the closure of X {\displaystyle X} can be constructed as the intersection of all of these closed supersets.
1. The strong finite intersection property says that the intersection of any finite number of elements of a set is infinite 2. A strong cardinal is a cardinal κ such that if λ is any ordinal, there is an elementary embedding with critical point κ from the universe into a transitive inner model containing all elements of V λ 3.