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 U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of the space there are sets U 1 ∈ U 1 , U 2 ∈ U 2 , … {\displaystyle U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\ldots } such that the family { U n : n ∈ N } {\displaystyle \{U_{n}:n\in \mathbb {N} \}} covers the space.
History In 1938, Fritz Rothberger introduced his property known as C ″ {\displaystyle C''} .
Characterizations
Combinatorial characterization For subsets of the real line, the Rothberger property can be characterized using continuous functions into the Baire space N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} . A subset A {\displaystyle A} of N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} is guessable if there is a function g ∈ A {\displaystyle g\in A} such that the sets { n : f ( n ) = g ( n ) } {\displaystyle \{n:f(n)=g(n)\}} are infinite for all functions f ∈ A {\displaystyle f\in A} . 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 c o v ( M ) {\displaystyle \mathrm {cov} ({\mathcal {M}})} is Rothberger.
Topological game characterization Let X {\displaystyle X} be a topological space. The Rothberger game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} played on X {\displaystyle X} is a topological game with two players Alice and Bob. 1st round: Alice chooses an open cover U 1 {\displaystyle {\mathcal {U}}_{1}} of X {\displaystyle X} . Bob chooses a set U 1 ∈ U 1 {\displaystyle U_{1}\in {\mathcal {U}}_{1}} . 2nd round: Alice chooses an open cover U 2 {\displaystyle {\mathcal {U}}_{2}} of X {\displaystyle X} . Bob chooses a set U 2 ∈ U 2 {\displaystyle U_{2}\in {\mathcal {U}}_{2}} . etc. If the family { U n : n ∈ N } {\displaystyle \{U_{n}:n\in \mathbb {N} \}} is a cover of the space X {\displaystyle X} , then Bob wins the game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} . Otherwise, Alice wins. A player has a winning strategy if he knows how to play in order to win the game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} (formally, a winning strategy is a function).
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