Selection principle

In mathematics, a selection principle is a rule assertingthe possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principlesstudies these principles and their relations to other mathematical properties.Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property. In mathematics, a selection principle is a rule assertingthe possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principlesstudies these principles and their relations to other mathematical properties.Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property. In 1924, Karl Menger introduced the following basis property for metric spaces: Every basis of the topology contains a sequence of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz observed that Menger's basis property is equivalent to the following selective property: for every sequence of open covers of the space, one can select finitely many open sets from each cover in the sequence, such that the selected sets cover the space.Topological spaces having this covering property are called Menger spaces. Hurewicz's reformulation of Menger's property was the first important topological property described by a selection principle. Let A {displaystyle mathbf {A} } and B {displaystyle mathbf {B} } be classes of mathematical objects.In 1996, Marion Scheepers introduced the following selection hypotheses,capturing a large number of classic mathematical properties: In the case where the classes A {displaystyle mathbf {A} } and B {displaystyle mathbf {B} } consist of covers of some ambient space, Scheepers also introduced the following selection principle. Later, Boaz Tsaban identified the prevalence of the following related principle: The notions thus defined are selection principles. An instantiation of a selection principle, by considering specific classes A {displaystyle mathbf {A} } and B {displaystyle mathbf {B} } , gives a selection (or: selective) property. However, these terminologies are used interchangeably in the literature. For a set A ⊂ X {displaystyle Asubset X} and a family F {displaystyle {mathcal {F}}} of subsets of X {displaystyle X} , the star of A {displaystyle A} in F {displaystyle {mathcal {F}}} is the set St ( A , F ) = ⋃ { F ∈ F : A ∩ F ≠ ∅ } {displaystyle { ext{St}}(A,{mathcal {F}})=igcup {Fin {mathcal {F}}:Acap F eq emptyset }} . In 1999, Ljubisa D.R. Kocinac introduced the following star selection principles: