Rothberger space

In mathematics, a Rothberger space is a topological space that satisfies a certain a basic selection principle. A Rothberger space is a space in which for every sequence of open covers formula_1 of the space there are sets formula_2 such that the family formula_3 covers the space.
History.
In 1938, Fritz Rothberger introduced his property known as formula_4.
Characterizations.
Combinatorial characterization.
For subsets of the real line, the Rothberger property can be characterized using continuous functions into the Baire space formula_5. A subset formula_6 of formula_5 is guessable if there is a function formula_8 such that the sets formula_9 are infinite for all functions formula_10. A subset of the real line is Rothberger iff every continuous image of that space into the Baire space is guessable. In particular, every subset of the real line of cardinality less than formula_11 is Rothberger.
Topological game characterization.
Let formula_12 be a topological space. The Rothberger game formula_13 played on formula_12 is a game with two players Alice and Bob.
1st round: Alice chooses an open cover formula_15 of formula_12. Bob chooses a set formula_17.
2nd round: Alice chooses an open cover formula_18 of formula_12. Bob chooses a finite set formula_20.
etc.
If the family formula_3 is a cover of the space formula_12, then Bob wins the game formula_13. Otherwise, Alice wins.
A player has a winning strategy if he knows how to play in order to win the game formula_13 (formally, a winning strategy is a function).