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 formula_1 and formula_2 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 formula_1 and formula_2 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 formula_1 and formula_2, gives a "selection (or: selective) property". However, these terminologies are used interchangeably in the literature.
Variations.
For a set formula_25 and a family formula_26 of subsets of formula_27, the star of formula_28 in formula_26 is the set formula_30.
In 1999, Ljubisa D.R. Kocinac introduced the following "star selection principles":
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 formula_27 be a topological space. An "open cover" of formula_27 is a family of open sets whose union is the entire space formula_43 For technical reasons, we also request that the entire space formula_27 is not a member of the cover. The class of open covers of the space formula_27 is denoted by formula_46. (Formally, formula_47, but usually the space formula_27 is fixed in the background.) The above-mentioned property of Menger is, thus, formula_49. In 1942, Fritz Rothberger considered Borel's strong measure zero sets, and introduced a topological variation later called Rothberger space (also known as "Cformula_50 space"). In the notation of selections, Rothberger's property is the property formula_51.
An open cover formula_52 of formula_27 is point-cofinite if it has infinitely many elements, and every point formula_54 belongs to all but finitely many sets formula_55. (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 formula_56-covers.) The class of point-cofinite open covers of formula_27 is denoted by formula_58. A topological space is a Hurewicz space if it satisfies formula_59.
An open cover formula_52 of formula_27 is an formula_62-cover if every finite subset of formula_27 is contained in some member of formula_52. The class of formula_62-covers of formula_27 is denoted by formula_67. A topological space is a γ-space if it satisfies formula_68.
By using star selection hypotheses one obtains properties such as star-Menger (formula_69), star-Rothberger (formula_70) and star-Hurewicz (formula_71).
The Scheepers Diagram.
There are 36 selection properties of the form formula_72, for formula_73 and formula_74. Some of them are trivial (hold for all spaces, or fail for all spaces). 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 formula_75 be a topological space, and formula_76. The class of sets formula_28 in the space formula_75 that have the point formula_79 in their closure is denoted by formula_80. The class formula_81 consists of the "countable" elements of the class formula_80. The class of sequences in formula_75 that converge to formula_79 is denoted by formula_85.
Topological Games.
There are close connections between selection principles and Topological Games.
The Menger game.
Let formula_27 be a topological space. The Menger game formula_102 played on formula_27 is a game for two players, Alice and Bob. It has an inning per each natural number formula_104. At the formula_105 inning, Alice chooses an open cover formula_106 of formula_27,
and Bob chooses a finite subset formula_108 of formula_52. 
If the family formula_110 is a cover of the space formula_27, 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 formula_127, but the selection principle formula_128 fails.
Examples and properties.
Subsets of the real line formula_135 (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 formula_136. For functions formula_137, write formula_138 if formula_139 for all but finitely many natural numbers formula_140. Let formula_28 be a subset of formula_136. The set formula_143 is bounded if there is a function formula_144 such that formula_145 for all functions formula_146. The set formula_143 is dominating if for each function formula_148 there is a function formula_149 such that formula_145.
Connections with other fields.
General topology.
Let P be a property of spaces. A space formula_27 is productively P if, for each space formula_75 with property P, the product space formula_156 has property P.
Function spaces.
Let formula_27 be a Tychonoff space, and formula_165 be the space of continuous functions formula_166 with pointwise convergence topology. 