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 and be classes of mathematical objects. In 1996, Marion Scheepers introduced the following selection hypotheses, capturing a large number of classic mathematical properties:
: For every sequence of elements from the class , there are elements such that . : For every sequence of elements from the class , there are finite subsets such that .
In the case where the classes and consist of covers of some ambient space, Scheepers also introduced the following selection principle.
: For every sequence of elements from the class , none containing a finite subcover, there are finite subsets such that .
Later, Boaz Tsaban identified the prevalence of the following related principle: : Every member of the class includes a member of the class .
The notions thus defined are selection principles. An instantiation of a selection principle, by considering specific classes and , gives a selection (or: selective) property. However, these terminologies are used interchangeably in the literature.
Variations
For a set and a family of subsets of , the star of in is the set .
In 1999, Ljubisa D.R. Kocinac introduced the following star selection principles:
: For every sequence of elements from the class , there are elements such that . : For every sequence of elements from the class , there are finite subsets such that .
The star selection principles are special cases of the general selection principles. This can be seen by modifying the definition of the family accordingly.
Covering properties
Covering properties form the kernel of the theory of selection principles. Selection properties that are not covering properties are often studied by using implications to and from selective covering properties of related spaces.
Let be a topological space. An open cover of is a family of open sets whose union is the entire space For technical reasons, we also request that the entire space is not a member of the cover. The class of open covers of the space is denoted by . (Formally, , but usually the space is fixed in the background.) The above-mentioned property of Menger is, thus, . In 1942, Fritz Rothberger considered Borel's strong measure zero sets, and introduced a topological variation later called Rothberger space (also known as C space). In the notation of selections, Rothberger's property is the property .
An open cover of is point-cofinite if it has infinitely many elements, and every point belongs to all but finitely many sets . (This type of cover was considered by Gerlits and Nagy, in the third item of a certain list in their paper. The list was enumerated by Greek letters, and thus these covers are often called -covers.) The class of point-cofinite open covers of is denoted by . A topological space is a Hurewicz space if it satisfies .
An open cover of is an -cover if every finite subset of is contained in some member of . The class of -covers of is denoted by . A topological space is a γ-space if it satisfies .
By using star selection hypotheses one obtains properties such as star-Menger (), star-Rothberger () and star-Hurewicz ().