On $(\mathbb{R}, \tau)$ the euclidean space of real numbers, we define a new topology by letting $\tau^{*}=\{X\subseteq \mathbb{R}: X=\emptyset \hspace{0.1cm}\mbox{or}\hspace{0.1cm}\mathbb{R}\setminus X\hspace{0.1cm}\mbox{is}\hspace{0.1cm} \mbox{compact} \hspace{0.1cm} \mbox{in}\hspace{0.1cm} (\mathbb{R}, \tau) \}$, it is known that $(\mathbb{R}, \tau^{*})$ is Lindelöf, meager in itself, my question is if Player II has a winning strategy in the Rothberger game played in $(\mathbb{R}, \tau^{*})$.

Remember that:

The Rothberger game on a topological space $X$ is played according to the following rules:

In each inning $n\in\omega$, Player I chooses an open cover $\mathcal{U_n}$ of $X$, and then Player II picks an open set $U_{n}\in\mathcal{U}_{n}$. At the end of the play $\langle \mathcal{U}_{0}, U_{0}, \mathcal{U}_{1}, U_{1}, ..., \mathcal{U}_{n}, U_{n}, .... \rangle $. The winner is Player II if $X\subseteq\bigcup_{n\in\omega}U_n$, and Player I otherwise.

Thank you

  • 1
    $\begingroup$ Crosspost from MSE $\endgroup$ – YuiTo Cheng Apr 24 '19 at 6:38
  • 2
    $\begingroup$ Crossposted too soon. This question has already been answered on math.SE. Player I has a winning strategy, not Player II. $\endgroup$ – bof Apr 24 '19 at 10:05
  • $\begingroup$ Since the question has been crossposted from math.SE, should I crosspost the easy answer from math.SE too? $\endgroup$ – bof Apr 24 '19 at 10:07
  • $\begingroup$ @bof I think it is more consistent with the trends/customs to close this question and avoid unnecessary duplication of content $\endgroup$ – Yemon Choi Apr 24 '19 at 14:27
  • $\begingroup$ I'm voting to close this question for the reason given in my previous comment $\endgroup$ – Yemon Choi Apr 24 '19 at 14:27

The game is a win for Player I, not Player II. A winning strategy for Player I is to choose, in inning $n$, the open cover $$\mathcal U_n=\{(a,b)\cup(-\infty,-10)\cup(10,\infty):a,b\in\mathbb R,\ 0\lt b-a\lt2^{-n}\}.$$ If more details are wanted, see the answer I posted to the same question at math.SE:



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