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 of the space there are sets such that the family covers the space.

History

In 1938, Fritz Rothberger introduced his property known as.

Characterizations

Combinatorial characterization

For subsets of the real line, the Rothberger property can be characterized using continuous functions into the Baire space. A subset of is guessable if there is a function such that the sets are infinite for all functions. 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 Cichoń's diagram| is Rothberger.

Topological game characterization

Let be a topological space. The Rothberger game played on is a game with two players Alice and Bob.
1st round: Alice chooses an open cover of. Bob chooses a set.
2nd round: Alice chooses an open cover of. Bob chooses a finite set.
etc.
If the family is a cover of the space, then Bob wins the game. Otherwise, Alice wins.
A player has a winning strategy if he knows how to play in order to win the game .