Timeline for Strategic vs. tactical closure
Current License: CC BY-SA 4.0
20 events
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| Dec 25, 2022 at 18:19 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 6 characters in body
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| Dec 25, 2022 at 16:24 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https
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| Dec 6, 2022 at 2:48 | comment | added | bof | @StevenClontz Thanks. The question, then, is whether the "Debs phenomenon" can be realized in an Alexandrov space. (I guess $T_0$/antisymmetry is irrelevant.) | |
| Dec 6, 2022 at 0:52 | comment | added | Steven Clontz | @bof I think that's right. Arbitrary intersections of open are open is known as Alexandrov, where open sets are characterized a pre-order. Toss in T0, and the pre-order is a partial order. | |
| Dec 6, 2022 at 0:38 | comment | added | bof | @MonroeEskew I'd been wondering how the poset BM game is related to the topological BM game, but it seems to me now that the former is a special case of the latter. Namely, if the poset is topologized by calling the down-sets open sets, then choosing a point $a\in P$ is tantamount to choosing the basic open set $\{x:x\le a\}$. So the poset BM game is just the topological BM game for $T_0$-spaces in which arbitrary intersections of open sets are open. Have I got that right? | |
| Dec 5, 2022 at 20:20 | comment | added | bof | @JoelDavidHamkins I never studied Debs' completely regular example, which seems to be rather complicated, but his first example was just the real line topologized with basic open sets of the form $(a,b)\setminus C$ where $C$ is countable. The winning strategy is easy; the proof of "no winning tactic", using the Baire category theorem, is trickier. | |
| Dec 5, 2022 at 2:52 | comment | added | Steven Clontz | I translated parts of the paper in grad school. I lost the original source I wrote but my advisor sent me a scan of some draft of this work a couple years back: github.com/StevenClontz/research/blob/master/… | |
| Dec 4, 2022 at 14:35 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
changed link to a search to the direct link to the review
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| Dec 4, 2022 at 11:05 | comment | added | Monroe Eskew | @bof I only asked one question. You’re raising a question of terminology. | |
| Dec 3, 2022 at 6:59 | comment | added | Monroe Eskew | @bof Set theorists use the term “Banach–Mazur game” for the poset game as I defined it. Maybe it is unjustified. | |
| Dec 2, 2022 at 22:39 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
Update in light of Will Brian's comment.
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| Dec 2, 2022 at 22:28 | comment | added | Monroe Eskew | @WillBrian Thanks for the explanation. So I will leave the question open for more answers. | |
| Dec 2, 2022 at 21:02 | comment | added | Joel David Hamkins | Ah, that sounds right. | |
| Dec 2, 2022 at 20:31 | comment | added | Will Brian | . . . but it does not answer the poset version. In fact, one of Debs' spaces (there are two -- one that's Hausdorff but non-regular, and a much more complicated one that is $T_{3 \frac{1}{2}}$) refines a metric space, and this means that player I can play in such a way that the sequence of moves is guaranteed to contain at most one point in the end, and therefore not contain a nonempty open set. | |
| Dec 2, 2022 at 20:29 | comment | added | Will Brian | Hi Joel, unfortunately, I don't think Debs' example answers Monroe's question. The problem is that the Banach-Mazur game on a topological space is not generally equivalent to the Banach-Mazur game on the poset consisting of its nonempty open sets. The problem is that in the topological version, II wins if there is a point contained in every set that's been played. In the poset version, II wins if there is a member of the poset included in every set that's been played. These are different conditions. Debs' space answers the topological-Banach-Mazur version of this question . . . (continued) | |
| Dec 2, 2022 at 19:33 | history | edited | Greg Martin | CC BY-SA 4.0 |
changed to inclusive pronouns
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| Dec 2, 2022 at 18:53 | vote | accept | Monroe Eskew | ||
| Dec 2, 2022 at 22:28 | |||||
| Dec 2, 2022 at 15:30 | comment | added | Joel David Hamkins | If someone could post an answer explaining the construction, I'd be grateful. | |
| Dec 2, 2022 at 15:25 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Dec 2, 2022 at 15:15 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |