Selection principle

In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets.

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 [1] 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[2] 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.

Hurewicz's reformulation of Menger's property was the first important topological property described by a selection principle.

In 1996, Marion Scheepers[3] introduced the following selection hypotheses, capturing a large number of classic mathematical properties: In the case where the classes

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

Covering properties form the kernel of the theory of selection principles.

is a family of open sets whose union is the entire space

For technical reasons, we also request that the entire space

The class of open covers of the space

In 1942, Fritz Rothberger considered Borel's strong measure zero sets, and introduced a topological variation later called Rothberger space (also known as C

(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

The class of point-cofinite open covers of

A topological space is a γ-space if it satisfies

By using star selection hypotheses one obtains properties such as star-Menger (

Restricting attention to Lindelöf spaces, the diagram below, known as the Scheepers Diagram,[3][5] presents nontrivial selection properties of the above form, and every nontrivial selection property is equivalent to one in the diagram.

Selection principles also capture important local properties.

There are close connections between selection principles and topological games.

inning, Alice chooses an open cover

Note that among Lindelöf spaces, metrizable is equivalent to regular and second-countable, and so the previous result may alternatively be obtained by considering limited information strategies.

[8] A Markov strategy is one that only uses the most recent move of the opponent and the current round number.

In a similar way, we define games for other selection principles from the given Scheepers Diagram.

In all these cases a topological space has a property from the Scheepers Diagram if and only if Alice has no winning strategy in the corresponding game.

Francis Jordan demonstrated a space where the selection principle

holds, but Alice has a winning strategy for the game

(with the induced subspace topology) holding selection principle properties, most notably Menger and Hurewicz spaces, can be characterized by their continuous images in the Baire space