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 of the space there are finite sets such that the family 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 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.
For functions, write if for all but finitely many natural numbers. A subset of is dominating if for each function there is a function such that. 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 is Menger.
The cardinality of Bartoszyński and Tsaban's counter-example to Menger's conjecture is

Properties