Pseudo-intersection

Pseudo-intersection

In the field of set theory, a branch of mathematical logic, pseudo-intersection is a concept that is used to describe a certain type of relationship between sets.

Definition

Given a collection of sets, a pseudo-intersection is a set that is almost contained in every set of the collection. More formally, if A is a collection of sets, a set B is a pseudo-intersection of A if for every set S in A, the set B is almost contained in S. This means that the difference between S and B is small in some sense.

Properties

Pseudo-intersections have several interesting properties. For example, if a collection of sets has a pseudo-intersection, then it also has an infimum, which is the greatest lower bound of the collection. This property can be used to prove various results in set theory and other areas of mathematics.

Applications

Pseudo-intersections are used in various areas of mathematics, including topology, measure theory, and functional analysis. They are particularly useful in the study of compact spaces, where they can be used to prove important results such as the Tychonoff's theorem.

See also

• Intersection (set theory)
• Union (set theory)
• Set (mathematics)