Hurewicz space

In mathematics, a Hurewicz space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Hurewicz space is a space in which for every sequence of open covers formula_1 of the space there are finite sets formula_2 such that every point of the space belongs to all but finitely many sets formula_3 .
History.
In 1926, Witold Hurewicz introduced the above property of topological spaces that is formally stronger than the Menger property. He didn't know whether Menger's conjecture is true, and whether his property is strictly stronger than the Menger property, but he conjectured that in the class of metric spaces his property is equivalent to formula_4-compactness.
Hurewicz's conjecture.
Hurewicz conjectured that in ZFC every Hurewicz metric space is σ-compact. Just, Miller, Scheepers, and Szeptycki proved that Hurewicz's conjecture is false, by showing that there is, in ZFC, a set of real numbers that is Menger but not σ-compact. Their proof was dichotomic, and the set witnessing the failure of the conjecture heavily depends on whether a certain (undecidable) axiom holds or not.
Bartoszyński and Shelah (see also Tsaban's solution based on their work ) gave a uniform ZFC example of a Hurewicz subset of the real line that is not σ-compact.
Hurewicz's problem.
Hurewicz asked whether in ZFC his property is strictly stronger than the Menger property. In 2002, Chaber and Pol in unpublished note, using dichotomy proof, showed that there is a Hurewicz subset of the real line that is not Menger. In 2008, Tsaban and Zdomskyy gave a uniform example of a Hurewicz subset of the real line that is Menger but not Hurewicz.
Characterizations.
Combinatorial characterization.
For subsets of the real line, the Hurewicz property can be characterized using continuous functions into the Baire space formula_5. For functions formula_6, write formula_7 if formula_8 for all but finitely many natural numbers formula_9. A subset formula_10 of formula_5 is bounded if there is a function formula_12such that formula_13 for all functions formula_14. A subset of formula_5 is unbounded if it is not bounded. Hurewicz proved that a subset of the real line is Hurewicz iff every continuous image of that space into the Baire space is unbounded. In particular, every subset of the real line of cardinality less than the bounding number formula_16 is Hurewicz.
Topological game characterization.
Let formula_17 be a topological space. The Hurewicz game played on formula_17 is a game with two players Alice and Bob.
1st round: Alice chooses an open cover formula_19 of formula_17. Bob chooses a finite set formula_21.
2nd round: Alice chooses an open cover formula_22 of formula_17. Bob chooses a finite set formula_24.
etc.
If every point of the space formula_17 belongs to all but finitely many sets formula_3 , then Bob wins the Hurewicz game. Otherwise, Alice wins.
A player has a winning strategy if he knows how to play in order to win the game (formally, a winning strategy is a function).
A topological space is Hurewicz iff Alice has no winning strategy in the Hurewicz game played on this space.
formula_27-neighborhood characterization.
A Tychonoff space formula_17 is Hurewicz iff for every compact space formula_29 containing the space formula_17, and a formula_27 subset G of formula_29 containing the space formula_17, there is a formula_4-compact set formula_35 with formula_36.