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I put a large bounty on this question to get a full detailed answer or to at least to draw some attention and get it up-voted. The bounty ends tomorrow and I got none.

What should I do? I feel maybe the question belongs to MO.

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    $\begingroup$ One problem is that the question is awfully broad. Another is that a lot of the kinds of topologies that you talk about (minimal/maximal with property $P$) don’t in general exist, so it’s very hard to know what sorts of lattice or partial order topologies on $\mathscr{T}$ or $\mathscr{T}/\sim$ might be at all helpful. I did just now add a couple of references on minimal topologies of various sorts and a pointer to some topologies on partial orders that have been studied. $\endgroup$ – Brian M. Scott Dec 16 '13 at 10:09
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That can certainly be frustrating, and is a risk any time you offer a bounty. I don't know what your background is, but please note that MathOverflow is intended primarily for professional mathematicians and graduate students doing cutting-edge research. A sufficiently interesting question from outside that community will sometimes get a useful answer, but it may also be closed, ignored, dismissed, or answered in a way that the person asking will not understand. Have you looked into the sources people suggested in the comments to your post here? I don't know if they will lead you to an answer, but you should probably have a decent understanding of how they relate before you head over to MO.

Final thoughts:

  1. Don't give up on getting an answer to a question with a bounty until the bitter end—bounty hunters may focus their attention on short-dated bounties.

  2. In the future, consider starting with a smaller bounty. This will allow you to start additional bounties after the first runs out, keeping your question in the spotlight longer.

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If you revise the question as Sergio suggested, you might use the occasion to modify the wording, currently in the fifth paragraph, that seems to refer to $\mathcal T(X)/\sim$ as a lattice. Although $\mathcal T(X)$ is a latice, I don't think $\mathcal T(X)/\sim$ is one.

Also, I don't think this question would last long on MO. I'd vote to close it as off-topic, and there are plenty of people there who are more ferocious than I am about closing questions.

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Honestly, your question is poorly written. I myself also write poor questions. They are long questions that do not make what is being asked clear from the start. My suggestion, make it shorter. Remove everything that is not essential. Be direct.

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