In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principles studies 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.
The main selection principles 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 family of all selected sets covers 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:
S 1 ( A , B ) {\displaystyle {\text{S}}_{1}(\mathbf {A} ,\mathbf {B} )} : For every sequence U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of elements from the class A {\displaystyle \mathbf {A} } , there are elements U 1 ∈ U 1 , U 2 ∈ U 2 , … {\displaystyle U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\dots } such that { U n : n ∈ N } ∈ B {\displaystyle \{U_{n}:n\in \mathbb {N} \}\in \mathbf {B} } .
S fin ( A , B ) {\displaystyle {\text{S}}_{\text{fin}}(\mathbf {A} ,\mathbf {B} )} : For every sequence U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of elements from the class A {\displaystyle \mathbf {A} } , there are finite subsets F 1 ⊆ U 1 , F 2 ⊆ U 2 , … {\displaystyle {\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots } such that ⋃ n = 1 ∞ F n ∈ B {\displaystyle \bigcup _{n=1}^{\infty }{\mathcal {F}}_{n}\in \mathbf {B} } . 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.
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