Rothberger space

For subsets of the real line, the Rothberger property can be characterized using continuous functions into the Baire space ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$. A subset ${\displaystyle A}$ of ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$ is guessable if there is a function ${\displaystyle g\in A}$ 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.

Let ${\displaystyle X}$ be a topological space. The Rothberger game ${\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )}$ played on ${\displaystyle X}$ is a game with two players Alice and Bob.

1st round: Alice chooses an open cover ${\displaystyle {\mathcal {U}}_{1}}$ of ${\displaystyle X}$. Bob chooses a set ${\displaystyle U_{1}\in {\mathcal {U}}_{1}}$.

2nd round: Alice chooses an open cover ${\displaystyle {\mathcal {U}}_{2}}$ of ${\displaystyle X}$. Bob chooses a 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 ${\displaystyle X}$, 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 ${\displaystyle X}$ be a metric space. Bob has a winning strategy in the game ${\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )}$ played on the space ${\displaystyle X}$ iff the space ${\displaystyle X}$ is countable.[3][4][5]