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 of the space there are finite sets such that every point of the space belongs to all but finitely many sets .

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 -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 axiom holds or not.
Bartoszyński and Shelah 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 In 2002, Chaber and Pol in unpublished note, using dichotomy proof, showed that there is a Hurewicz subset of the real line that is not Menger.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. For functions, write if for all but finitely many natural numbers. A subset of is bounded if there is a function such that for all functions. A subset of 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 is Hurewicz.

Topological game characterization

Let be a topological space. The Hurewicz game played on is a game with two players Alice and Bob.
1st round: Alice chooses an open cover of. Bob chooses a finite set.
2nd round: Alice chooses an open cover of. Bob chooses a finite set.
etc.
If every point of the space belongs to all but finitely many sets , 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.
A topological space is Hurewicz iff Alice has no winning strategy in the Hurewicz game played on this space.

G_\delta-neighborhood characterization

A Tychonoff space is Hurewicz iff for every compact space containing the space, and a subset G of containing the space, there is a -compact set with.

Properties