Timeline for Potential connected non-Lie subgroup
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2009 at 20:17 | history | edited | Martin M. W. | CC BY-SA 2.5 |
added 21 characters in body
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| Oct 30, 2009 at 18:34 | vote | accept | David E Speyer | ||
| Oct 30, 2009 at 17:53 | comment | added | Martin M. W. | Yeah, my temples are throbbing now, too! I think the key is we only have to deal with "candidate" open sets where the projection of the boundary onto the x-axis contains an interval. That's because if the graph G = A union B for open sets A,B, then the projections proj(A) and proj(B) onto the x-axis are both open, and must intersect. The set proj(A) intersect proj(B) contains an interval, and for any x in this interval, there is a y such that (x,y) is in boundary(A). I've edited the answer to take this point into account. | |
| Oct 30, 2009 at 17:52 | history | edited | Martin M. W. | CC BY-SA 2.5 |
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| Oct 30, 2009 at 15:34 | comment | added | David E Speyer | "[U]nless the open set is the union of vertical lines[,] the projection of the boundary contains an interval." Is this clear? That's exactly the sort of issue that made my head hurt when I was thinking about this. | |
| Oct 30, 2009 at 13:48 | comment | added | Martin M. W. | Good point! But I think this doesn't kill the argument. We actually only have to worry about hitting the boundaries of open sets whose complement contains an open set. (since we're trying to avoid the case where graph contained in union of two disjoint open sets.) For these sets the construction still goes through, because unless the open set is the union of vertical lines the projection of the boundary contains an interval. | |
| Oct 30, 2009 at 13:18 | comment | added | David E Speyer | Wait a sec, something is wrong with this argument. For any point (x,y) in R^2, the set R^2 \setminus {(x,y)} is open, with boundary {(x,y)}. So a graph which meets every boundary would have to be all of R^2. | |
| Oct 30, 2009 at 13:12 | history | edited | Martin M. W. | CC BY-SA 2.5 |
explicitly dealt with open sets that are unions of vertical lines.
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| Oct 30, 2009 at 12:48 | history | edited | David E Speyer | CC BY-SA 2.5 |
Excluded the trivial open sets
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| Oct 29, 2009 at 23:06 | vote | accept | David E Speyer | ||
| Oct 30, 2009 at 13:17 | |||||
| Oct 29, 2009 at 21:36 | history | edited | Martin M. W. | CC BY-SA 2.5 |
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| Oct 29, 2009 at 20:54 | history | answered | Martin M. W. | CC BY-SA 2.5 |