Selection principle

In mathematics, the theory of selection principles deals with the possibility of obtaining mathematically significant objects by selecting elements from sequences of sets. The properties studied mainly include covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially functions 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 ${\displaystyle \mathbf {A} }$ and ${\displaystyle \mathbf {B} }$ be classes of mathematical objects. In 1996, Marion Scheepers introduced the following selection hypotheses, capturing a large number of classic mathematical properties:

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