Let $X$ be a topological space. The point-open game $G_{po}(X)$ is defined as folows. It is played by two players ONE and TWO. In the n'th step $(n \in \omega)$, ONE choose a finite subset $F$ of $X$, and TWO selects an open $G_n$ in $X$, $F_n \subset G_n$. ONE wins if $\bigcup \{ G_n : n \in \omega \} = X$, otherwise TWO wins.


If $\langle A_n : n \in \omega \rangle$ is a sequence of subsets of a set $X$, $$ \underline{Lim} A_n = \{ x \in X : \exists n_0 \in \omega \forall n \geq n_0, x \in A_n \} $$ If $\mathcal A$ is a family of subsets of a set $X$, then, $L(\mathcal A)$ denotes the smallest family of subsets of $X$ containing $\mathcal A$ and closed under $\underline{Lim}$.

I am trying to prove that (a)$\Rightarrow$(b) where:

(a) If $\mathcal I$ is an open $\omega$-cover of $X$, then, there is a sequence $G_n \in \mathcal I$, with $\underline{Lim} G_n = X$.

(b) If $\mathcal I$ is an open $\omega$-cover of $X$, then $X \in L(\mathcal I)$.

A family $\mathcal A$ of subsets of a set $A$ is said to be an $\omega$-cover of $X$, if for any finite subset $F$ of $X$,, there is an $A \in \mathcal A$ with $F \subset A$.

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    $\begingroup$ A couple questions. (1) What does your question have to do with the point-open game? (Your question seems to only involve $\underline{\mathrm{Lim}}$, $L(\mathcal{I})$, and open $\omega$-covers.) (2) Isn't the implication trivial? If (a) holds, then given any open $\omega$-cover $\mathcal{I}$ of $X$ there is a sequence $\langle G_n \rangle_n$ in $\mathcal{I}$ such that $\underline{\mathrm{Lim}}_n G_n = X$. Since $G_n \in L(\mathcal{I})$ for each $n$ and $L(\mathcal{I})$ is closed under the $\underline{\mathrm{Lim}}$ operation, it must be that $X \in L(\mathcal{I})$. $\endgroup$ – user642796 Apr 30 '14 at 9:06
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    $\begingroup$ Also, I think what you're calling the "point-open" game should be called the "finite-open" game; it would be the "point-open" game if the finite sets $F_n$ were required to be singletons. (Although the two games do seem to be more or less equivalent.) $\endgroup$ – bof Apr 30 '14 at 9:14
  • $\begingroup$ The Claim I stated is a part of more general proposition which incudes also point-open game.. It is in the bottom of page 153 of this article: ac.els-cdn.com/0166864182900657/… Also, the definition for point open game is from that article. Anyhow, I see now, that it is trivial. I was missing the part of "Closed under Lim". Thank you both!! $\endgroup$ – topsi Apr 30 '14 at 10:20

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