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2h

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8h

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Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?
Crossposting is frowned upon. This is a site for research questions of professional mathematicians; it should be considered a niche site. I recommend that you edit the MSE question to include more detail as you have here, and wait a little longer. The tags you have applied here are appropriate and should be used there as well, if they exist there. If you don't get an answer in a week, then you can try here, but some users may feel the level isn't right for this site. 
9h

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What is the most useful nonexisting object of your field?
@l0b0 If one were being fully formal, then yes, that would have to be specified. But in a context of people talking who can follow the reference implied by the technical terms "the elliptic curve attached to...", it's tacitly understood; it goes without saying. :) 
21h

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What is the *smallest* number used in a mathematical paper?
What the heck is the nt tag? Also, I think it might be fitting if we had an 'idle curiosity' tag. :) 
1d

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What is the most useful nonexisting object of your field?
Yemon, thanks for your understanding. I'm removing my first comment and I'm voting up. 
1d

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The impact of large cardinals in mathematics
Changes in question formulation are often frustrating, because they often nullify the relevance of earlier answers. Jeff Strom's answer, which I thought was good, now doesn't seem very relevant to any of Q14 at present. Having said that: why restrict to finite and discrete mathematics in Q4? Also, aren't Q2 and Q3 pretty similar now? Couldn't they be combined? 
1d

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The impact of large cardinals in mathematics
"Most impact" refers to an earlier version of the question, now edited out. This seems like a reasonable answer. 
1d

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Are there proofs that you feel you did not “understand” for a long time?
@StevenGubkin Here's a pretty good gif, if you want to explain this to students one day: commons.wikimedia.org/wiki/File:ExpIPi.gif. 
1d

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Are there proofs that you feel you did not “understand” for a long time?
@StevenGubkin Yes, precisely! In terms of pictures, successive powers of $1 + \frac{ix}{n}$ make me think of a growing snail shell that for large $n$ hews fairly close to a circle. Once upon a time people thought of $e^{i\pi} + 1 = 0$ as something almost mystical, but such insights dispel the mystery and make it seem close to obvious. I still think it's wonderful! 
1d

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What is the most useful nonexisting object of your field?
Neither is $\mathbb{F}_1$, but if we are liberal and admit such chimerical entities, then both are worthy of mention! 
1d

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A. Markov's papers?
I'd like to mention that an Area 51 StackExchange site on History of Science and Mathematics is in the middle of a Commitment stage: area51.stackexchange.com/proposals/65204/… 
1d

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What is the most useful nonexisting object of your field?
Maybe it should be added that free suplattices on any set exist, and suplattices admit arbitrary infs (so are complete). Here, the morphisms of $\mathbf{SupLat}$ preserve just sups. Similarly, the free inflattice exists on any set, and this admits arbitrary sups. It's when we require morphisms of the category to preserve both arbitrary sups and infs that free objects do not exist (on sets of cardinaility greater than 2). 
1d

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What is the most useful nonexisting object of your field?
This is about the only example I've seen so far that actually has been given a special name (see the opening paragraph of the OP): it's called the Frey curve. (Edit: well, now there are others, such as Reinhardt cardinal.) 
2d

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What is the most useful nonexisting object of your field?
Since the phrase "proof by contradiction" was invoked, I'll recall the distinction between "proof of negation" and "proof by contradiction", as explained by Andrej Bauer here: math.andrej.com/2010/03/29/…. For proofs of negation, one supposes a proposition $\phi$ is true and derives a contradiction; therefore $\phi$ is false. A proof by contradiction supposes $\phi$ is false and derives a contradiction; therefore $\phi$ is true. Intuitionists accept proofs of negation, but not proof by contradiction! 
2d

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What is the most useful nonexisting object of your field?
Yemon, your followup definitely improves your answer. Answers should provide some context, rather than being baldly obviously nonexistent items. 
2d

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What is the most useful nonexisting object of your field?
@NAME_IN_CAPS Yes, very true; I like that response because it describes the thought in context. 
2d

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Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces
I agree that it is good MO form to start new threads rather than change the original question. I do observe that Tristan made a note of the form of the original question at the end, so that the relevance of Nik's answer can still be discerned, but if not now, let's please follow Ricardo's advice in the future. 
2d

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If any open set is a countable union of balls, does it imply separability?
Ah, very good, Joel  thanks for looking into this. I had begun recording these arguments in the nLab, and now I'm convinced. :) 
2d

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If any open set is a countable union of balls, does it imply separability?
Joel, I'm probably being thick, but why must a ball $B_s(x)$ that contains two $x_\alpha$, $x_\beta$ also exclude $p_\alpha$? 
Sep 26 
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Functoriality of the right Kan extension
Bubbles, that is not what you wrote. You wrote: "why must the choices determining the functor $Ran_L$ coincide... making $Ran_{LK}$ and $Ran_L Ran_K$ equal". I answered that they need not. If you now want to change the question to ask about a possibility and not a necessity, then please make an appropriate edit, or ask a new question altogether. 