Timeline for Strategic vs. tactical closure
Current License: CC BY-SA 4.0
9 events
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| Dec 6, 2022 at 1:01 | history | edited | Steven Clontz | CC BY-SA 4.0 |
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| Dec 5, 2022 at 16:50 | comment | added | Will Brian | @StevenClontz: Yes, you're right. Also, I posted my comment before taking the time to digest Joel's. I think a simple modification of Debs' space cannot answer Monroe's question, unless the modification somehow avoids the argument given in bof's answer. | |
| Dec 5, 2022 at 16:23 | comment | added | Steven Clontz | @Will on that poset as P2 I would immediately pick some $(\alpha,x)$ in the initial round. Then the set below me is a copy of $\omega_1$ and I believe I win regardless of the space. | |
| Dec 5, 2022 at 14:04 | comment | added | Will Brian | Steven, what if you use the poset on $\omega_1 \times (X \cup \mathcal T_X^{\neq \emptyset})$ where $(\alpha,U) \leq (\beta,V)$ if and only if $\alpha > \beta$ and $U \subseteq V$, and $(\alpha,x) \leq (\beta,V)$ if and only if $\alpha > \beta$ and $x \in V$, and $(\alpha,x) \leq (\beta,y)$ if and only if $\alpha > \beta$ and $x=y$? | |
| Dec 5, 2022 at 14:02 | comment | added | Will Brian | @JoelDavidHamkins: Debs' space does indeed have bof's property. This is essentially how he proves that II has a winning $2$-tactic in the BM game. | |
| Dec 5, 2022 at 13:11 | comment | added | Joel David Hamkins | I wonder whether the construction will fall prey to the argument of bof's answer? Namely, if Deb's space has the property that one can find a basis for the topology with only countably many basis elements in every upper cone, then there will be a winning tactic in the poset game. Your limit construction idea would seem to preserve that property. | |
| Dec 5, 2022 at 5:18 | comment | added | Steven Clontz | There's an error at the limit stage of my construction: there's no immediate successor to restrict the open sets to, so there's no way to force the opponent to pick subsets. So I think just sticking $\mathcal T_X\setminus \{\emptyset\}$ at each stage may work, but the rest of the argument needs to adjust for a "cheater" who doesn't choose subsets. | |
| Dec 5, 2022 at 5:07 | history | edited | Steven Clontz | CC BY-SA 4.0 |
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| Dec 5, 2022 at 4:52 | history | answered | Steven Clontz | CC BY-SA 4.0 |