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sniffnoyProblem from this week's algebra set:
We have a topological space; we define a locally closed set to be an intersection of an open set and a closed set, and a constructible set to be a finite union of locally closed sets. First part was to prove that constructible sets are closed under finite union (duh) and complementation, showing that in fact they're the same thing as what you might call "finite Borel".
Second part was to prove that any constructible set is in fact a finite disjoint union of locally closed sets. The upper bound here is for, if you've got a union of n locally closed sets, it's a disjoint union of f(n) locally closed sets. The problem of course just asked for finiteness, but the way I did it started me wondering about it.
Now it's pretty easy to get an upper bound of 7 for f(2), and 7 being finite, the natural thing to do is to induct and get that f(n) is in fact always finite. But I couldn't get the "obvious" induction to work, even for n=3, and in fact I'm pretty certain it can't (there's a disjointness problem), though Nic and some others said they did it that way (I haven't gone over it with them, so maybe their "obvious" induction is actually a bit different). So what I ended up doing was saying, if S is a union of n locally closed sets, its complement is a disjoint union of 3^n such, and so S is a disjoint union of 3^3^n such. Of course, 3^9 is far larger than 7, so clearly this is nowhere near strict. (Also, 27 is bigger than 1, and 3 is bigger than 0.)
I don't really intend to work on this problem right now, and it's probably not significant for anything, but just something I started thinking about. (Also, attempting to find *lower* bounds looks "yikes"ey[0]. Trying to find a topological space and a union of 2 locally closed sets that cannot be expressed as a disjoint union of less than 7 locally closed sets? Yikes.)
ADDENDUM: Also, apparently the fact that finitely generated rings are noetherian is actually useful, for generalizing results from the noetherian case. Huh. I didn't expect that.
-Harry
[0]Yes, that was the best way I could think of to say this.
We have a topological space; we define a locally closed set to be an intersection of an open set and a closed set, and a constructible set to be a finite union of locally closed sets. First part was to prove that constructible sets are closed under finite union (duh) and complementation, showing that in fact they're the same thing as what you might call "finite Borel".
Second part was to prove that any constructible set is in fact a finite disjoint union of locally closed sets. The upper bound here is for, if you've got a union of n locally closed sets, it's a disjoint union of f(n) locally closed sets. The problem of course just asked for finiteness, but the way I did it started me wondering about it.
Now it's pretty easy to get an upper bound of 7 for f(2), and 7 being finite, the natural thing to do is to induct and get that f(n) is in fact always finite. But I couldn't get the "obvious" induction to work, even for n=3, and in fact I'm pretty certain it can't (there's a disjointness problem), though Nic and some others said they did it that way (I haven't gone over it with them, so maybe their "obvious" induction is actually a bit different). So what I ended up doing was saying, if S is a union of n locally closed sets, its complement is a disjoint union of 3^n such, and so S is a disjoint union of 3^3^n such. Of course, 3^9 is far larger than 7, so clearly this is nowhere near strict. (Also, 27 is bigger than 1, and 3 is bigger than 0.)
I don't really intend to work on this problem right now, and it's probably not significant for anything, but just something I started thinking about. (Also, attempting to find *lower* bounds looks "yikes"ey[0]. Trying to find a topological space and a union of 2 locally closed sets that cannot be expressed as a disjoint union of less than 7 locally closed sets? Yikes.)
ADDENDUM: Also, apparently the fact that finitely generated rings are noetherian is actually useful, for generalizing results from the noetherian case. Huh. I didn't expect that.
-Harry
[0]Yes, that was the best way I could think of to say this.
no subject
Date: 2009-02-01 07:52 am (UTC)