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 \( {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } \) of the space there are sets \( {\displaystyle U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\ldots } \) such that the family \( {\displaystyle \{U_{n}:n\in \mathbb {N} \}} \) covers the space.
History
In 1938, Fritz Rothberger introduced his property known as \( {\displaystyle C''} \).[1]
Characterizations
Combinatorial characterization
For subsets of the real line, the Rothberger property can be characterized using continuous functions into the Baire space \( {\mathbb {N}}^{{\mathbb {N}}}\) . A subset A of \( {\mathbb {N}}^{{\mathbb {N}}} \) is guessable if there is a function \( g\in A \) such that the sets\( {\displaystyle \{n:f(n)=g(n)\}} \) are infinite for all functions \( {\displaystyle f\in A} \) . 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 \( {\displaystyle \mathrm {cov} ({\mathcal {M}})} \)[2] is Rothberger.
Topological game characterization
Let X be a topological space. The Rothberger game \( {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} \) played on X is a game with two players Alice and Bob.
1st round: Alice chooses an open cover \( {\displaystyle {\mathcal {U}}_{1}} \) of X. Bob chooses a set \( {\displaystyle U_{1}\in {\mathcal {U}}_{1}}. \)
2nd round: Alice chooses an open cover \({\displaystyle {\mathcal {U}}_{2}}\( of X. Bob chooses a finite set \( {\displaystyle U_{2}\in {\mathcal {U}}_{2}} \).
etc.
If the family\( {\displaystyle \{U_{n}:n\in \mathbb {N} \}} \) is a cover of the space X, then Bob wins the game \( {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} \). Otherwise, Alice wins.
A player has a winning strategy if he knows how to play in order to win the game \( {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} \)(formally, a winning strategy is a function).
A topological space is Rothberger iff Alice has no winning strategy in the game\( {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} \) played on this space.[3]
Let X be a metric space. Bob has a winning strategy in the game \( {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} \) played on the space X iff the space X is countable.[3][4][5]
Properties
Every countable topological space is Rothberger
Every Luzin set is Rothberger[1]
Every Rothberger subset of the real line has strong measure zero.[1]
In the Laver model for the consistency of the Borel conjecture every Rothberger subset of the real line is countable
References
Rothberger, Fritz (1938-01-01). "Eine Verschärfung der Eigenschaft C". Fundamenta Mathematicae (in German). 30 (1). ISSN 0016-2736.
Bartoszynski, Tomek; Judah, Haim (1995-08-15). Set Theory: On the Structure of the Real Line. Taylor & Francis. ISBN 9781568810447.
Pawlikowski, Janusz. "Undetermined sets of point-open games". Fundamenta Mathematicae. 144 (3). ISSN 0016-2736.
Scheepers, Marion (1995-01-01). "A direct proof of a theorem of Telgársky". Proceedings of the American Mathematical Society. 123 (11): 3483–3485. doi:10.1090/S0002-9939-1995-1273523-1. ISSN 0002-9939.
Telgársky, Rastislav (1984-06-01). "On games of Topsoe". Mathematica Scandinavica. 54: 170–176. doi:10.7146/math.scand.a-12050. ISSN 1903-1807.
vte
Topology
Fields
General (point-set) Algebraic Combinatorial Continuum Differential Geometric
low-dimensional Homology
cohomology Set-theoretic
Computer graphics rendering of a Klein bottle
Key concepts
Open set / Closed set Continuity Space
compact Hausdorff metric uniform Homotopy
homotopy group fundamental group Simplicial complex CW complex Manifold Second-countable space
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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