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Feature requests: "Do the empty set AND the entire set really need to be open"- Reasons for closing and downvoting?

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    Hi,

    I am referring to the "Do the empty set AND the entire set really need to be open?" thread (-> http://mathoverflow.net/questions/19243/do-the-empty-set-and-the-entire-set-really-need-to-be-open). I have seen there are already 3 votes to close, so I would like to ask what the reasons for these might be. Also, it seems there have been at least 3-4 downvotes, but no one cared to explain what for. As I see it, the thread had some interesting development, including the fact that it turns out that the 1st axiom is not merely of technical nature as some were saying in the beginning of the discussion. And the implications of omitting the 1st axiom seem (at least to me) not trivially pathological (i.e. such that we won´t be able to do any useful mathematics)...

    Thanks in advance,

    efq
    • CommentAuthorHarry Gindi
    • CommentTimeMar 25th 2010 edited
     
    It doesn't seem to me like you've actually put much thought into it. So now it has four votes to close.
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    I didn't downvote your question, but I can guess why people did: having the empty set and the entire set be open is a fundamental aspect of topology, and to remove those properties would be to remove fundamental properties that we want topologies to have. For example, a basic aspect of topological spaces is that finite intersections and arbitrary unions of open sets are open, but if the empty set and the entire space were not open then this property would not hold! Perhaps you would like to make up a strange new area of mathematics where finite intersections of open sets are no longer open, but people would object if you called it "topology."
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    Answer: Topology already has enough trivial counterexamples =p.
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    @Chuck Hague: There were similar comments in the thread as well. However, if one carefully reads my post, one should see that NOWHERE I forbid the entire set and the empty set to be open. I just drop the requirement for them to be so, which is quite different.

    @fpqc: Unfortunately, more than enough :-)
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    ex falso quodlibet, I voted to close, for the same reasons as Pete Clark. I don't think I can explain those reasons any better than he did.

    There have been other instances in which people have asked a question that was generally thought to be not very good, but interesting answers came in. The fact that interesting answers come in doesn't imply that it was a good question, though. It could just mean that there are lots of interesting people round here, who at the slightest prompt will write interesting things.

    Clearly it's disheartening when people vote your question down or vote to close. But it's certainly nothing personal: it's only a judgement on the question, not you.

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    By the way, the link to the page above reveals an interesting StackExchange bug -- you get a "lower level" server error message than you should. I'll include the link again here in case it gets fixed above.

    http://mathoverflow.net/questions/19243/do-the-empty-set-and-the-entire-set-really-need-to-be-open).
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    Presumably, it would also be useful to have the actual link to the question in case it doesn't get fixed above!

    http://mathoverflow.net/questions/19243/do-the-empty-set-and-the-entire-set-really-need-to-be-open

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