Menger space

In mathematics, a Menger space is a topological space that satisfies a certain a basic selection principle that generalizes σ-compactness. A Menger 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 the family formula_3 covers the space.
History.
In 1924, Karl Menger 
introduced the following basis property for metric spaces: 
Every basis of the topology contains a countable family of sets with vanishing 
diameters that covers the space. Soon thereafter, 
Witold Hurewicz 
observed that Menger's basis property can be reformulated to the above form using sequences of open covers.
Menger's conjecture.
Menger conjectured that in ZFC every Menger metric space is σ-compact. 
Fremlin and Miller 
proved that Menger's conjecture is false, by showing that there is,
in ZFC, a set of real numbers that is Menger but not σ-compact. 
The Fremlin-Miller 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 Tsaban
gave a uniform ZFC example of a Menger subset of the real line that is not σ-compact.
Combinatorial characterization.
For subsets of the real line, the Menger property can be characterized using continuous functions into the Baire space formula_4.
For functions formula_5, write formula_6 if formula_7 for all but finitely many natural numbers formula_8. A subset formula_9 of formula_4 is dominating if for each function formula_11 there is a function formula_12 such that formula_13. Hurewicz proved that a subset of the real line is Menger iff every continuous image of that space into the Baire space is not dominating. In particular, every subset of the real line of cardinality less than the dominating number formula_14 is Menger.
The cardinality of Bartoszyński and Tsaban's counter-example to Menger's conjecture is
formula_14.