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. 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 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 and 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 and 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 and, gives a selection 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:
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 . 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. 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. 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.

The Scheepers Diagram

There are 36 selection properties of the form, for and. Some of them are trivial. Restricting attention to Lindelöf spaces, the diagram below, known as the Scheepers Diagram, presents nontrivial selection properties of the above form, and every nontrivial selection property is equivalent to one in the diagram. Arrows denote implications.

Local properties

Selection principles also capture important non-covering properties.
Let be a topological space, and. The class of sets in the space that have the point in their closure is denoted by. The class consists of the countable elements of the class. The class of sequences in that converge to is denoted by.
There are close connections between selection principles and Topological Games.

The Menger game

Let be a topological space. The Menger game played on is a game for two players, Alice and Bob. It has an inning per each natural number. At the inning, Alice chooses an open cover of,
and Bob chooses a finite subset of.
If the family is a cover of the space, then Bob wins the game. Otherwise, Alice wins.
A strategy for a player is a function determining the move of the player, given the earlier moves of both players. A strategy for a player is a winning strategy if each play where this player sticks to this strategy is won by this player.
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. 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. But this does not hold in general; Francis Jordan demonstrated a space where Alice has a winning strategy for, but the selection principle fails.

Examples and properties

Subsets of the real line holding selection principle properties, most notably Menger and Hurewicz spaces, can be characterized by their continuous images in the Baire space. For functions, write if for all but finitely many natural numbers. Let be a subset of. The set is bounded if there is a function such that for all functions. The set is dominating if for each function there is a function such that.

General topology

Let P be a property of spaces. A space is productively P if, for each space with property P, the product space has property P.
Let be a Tychonoff space, and be the space of continuous functions with pointwise convergence topology.