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Leaky Nun
Leaky Nun
08:48
@BalarkaSen
Balarka Sen
Balarka Sen
@LeakyNun
Leaky Nun
Leaky Nun
@BalarkaSen do you have a construction of Top->Haus that avoids universe issues and transfinite recursion?
Balarka Sen
Balarka Sen
I do not
Leaky Nun
Leaky Nun
ok
Balarka Sen
Balarka Sen
The question is way out of my cup of tea
 
8 hours later…
Mike Miller
Mike Miller
16:26
@Leaky A Hausdorff space is defined by the property that every convergent net has a unique limit. I think given a cardinality bound on the space you can also come up with a cardinality bound on the nets you need to consider.
Then I think your Hausdorffification functor spits out the quotient space where you identify two points if they are the limit of the same net
MatheinBoulomenos
MatheinBoulomenos
16:47
@MikeMiller the only nets you need are indexed by neighborhoods of a point (but then you might just work with filters directly)
Mike Miller
Mike Miller
17:04
Does this resolve the problem though?
MatheinBoulomenos
MatheinBoulomenos
17:18
The construction that Leaky is referring to doesn't use nets or filters, so probably not
Mike Miller
Mike Miller
18:09
@MatheinBoulomenos isn't he looking for an alternate construction that doesn't have problems with unbounded size of sets involved?
I think that what I'm describing is a/the left adjoint to the inclusion Haus -> Top... and I thought that was what desired
MatheinBoulomenos
MatheinBoulomenos
18:26
yes that's true. This might work, but the details seem to be tricky for this one, as the fact that one possible construction uses transfinite induction shows
It seems like a good thing to try, but I'm not sure if it actually works until I see it. The construction with transfinite induction "looks like" one should be done after the first step
Mike Miller
Mike Miller
Fair enough
Mike Miller
Mike Miller
19:11
I see your point - I might just be reconstructing the transfinite induction proof
I identify nets, and then identify nets in the quotient space, and so on
But I wonder if you can combine these two ideas and say you can put a cardinality bound on the Hausdorff codomain you consider when checking if maps identify points
MatheinBoulomenos
MatheinBoulomenos
19:42
@MikeMiller the usual construction identifies points that can't be separated by disjoint neighborhoods (and then does the same on the quotient, and the quotient of that ...)
Mike Miller
Mike Miller
Yeah that's the same idea as mine I feel
But it seems that the second construction (identify points that get identified with all Hausdorff codomain) can be done just as well with surjective maps to Hausdorff codomain, right?
So you avoid universes
MatheinBoulomenos
MatheinBoulomenos
that seems right
yeah, I don't see why you would need non-surjective maps
Mike Miller
Mike Miller
Just restrict to the image
MatheinBoulomenos
MatheinBoulomenos
right
Mike Miller
Mike Miller
20:02
It's been a very long time since I thought about spaces that weren't exceptionally good
Hausdorff is a bare minimum
Alessandro Codenotti
Alessandro Codenotti
20:42
user image
I'll probably have to learn as much as I can from this list of topics over the summer, any book/notes suggestions?
My current knowledge of algebraic topology basically stops at Seifert-van Kampen
2
 
2 hours later…
Mike Miller
Mike Miller
22:40
Those are all good. It strikes me as "read Hatcher" though
MatheinBoulomenos
MatheinBoulomenos
23:23
You'll need something to sumplement Hatcher for DeRham, though

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