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http://mathoverflow.net/questions/38763/is-lp-mathbbr-minus-the-zero-function-contractible and http://mathoverflow.net/questions/198/how-do-you-show-that-s-infty-is-contractible both cover the same ground. Both got satisfactory answers, but it's interesting to note that the separation in time also led to a separation in users in that I was probably the first on the scene who knew of the earlier question.
Clearly, for posterity, they should be linked. I've put comments on both with links to the other.
If I'd been the first to see the second question, I would have voted to close as duplicate. Now, I'm a little hesitant to do so. I'm not completely sure why - there doesn't seem to be a good reason to not vote to close "as duplicate" since it's not saying it's a bad question in any way, but I'd rather find out what others think before deciding.
I guess I'm also thinking about precedents here (so saying, "Well, I think in this case it's okay" is not really satisfactory), is it better for the site if such situations still get closed as duplicates, or not?
@Andrew: just because one answer serves for two different questions doesn't mean the questions are duplicates (exact or otherwise)!
In this case I think that the questions should not be merged or closed. Your insight that they can both be answered in the same way is valuable, so your linking of each question to the other in the comments was a positive contribution. Thanks for doing so.
As an algebraic topologist, these questions really do feel so close as to be exact duplicates. The homotopy equivalence of a vector space minus one point and its sphere is one of those simple algebraic topology exercises that if a questioner doesn't know it, they're at too basic a level in algebraic topology for MO. So, to make an extremely awful joke, in the homotopy category of MO questions, these are equivalent questions. (An extremely interesting and important point is that actually these are equivalent at the higher level since the sphere is diffeomorphic to its ambient space in many cases, as Dick Palais says.)
In addition, I think that one could make the case that if one answer serves for two questions then those questions must be duplicates even if it isn't obvious! Of course, that line would probably be a bit extreme as a reason for closing, but it's an intriguing idea!
@Andrew: but isn't the topology on the ambient vector space different in the two questions? If it were the same, then yes, I would agree that the questions are essentially the same. As is, I don't see how to pass from one question to the other without knowing a certain answer that works for both. (Note that some people gave answers that do not work for both.)
I really don't buy the argument of your second paragraph. It is an important part of mathematics that the same (in some sense) proof can serve for many theorems.
Andy, let me briefly respond to that last point. I think that "closed as duplicate" is very different to all of the other reasons for closure because it doesn't say that the question is not suitable. It just says that it was asked before and, perhaps just from trying to be neat, only one should be left open. So it should be taken as, "Nice question, but we've seen it before and here's what was said last time.".
If you look carefully, I did not vote to close as duplicate, I started the discussion before doing so and since Tyler disagrees with me on the duplication and I know him to be well-versed in algebraic topology, that's enough for me to leave things as they are. In addition, if you look carefully, I took the opportunity to add to my original answer to widen its scope.
So please tell me what of that is "mean-spirited".
Andrew makes a good point here: one would hope that "closed as duplicate" carries less opprobrium than all the other reasons for closure. On the other hand, maybe that's too much to expect. I know in the earlier days of MathOverflow many of us hoped that "people wouldn't take closure personally", and eventually came round to the realisation that this just wasn't what happened. Do we have to be careful here? Are people going to be offended by having their question closed as a duplicate, even in the absolutely clear-cut situations? I hope not.
Re-reading my comment to Andy, I'd like to add that I'm aware that things come across differently to how they were intended - especially over the internet - so if any of my actions do come across as "mean-spirited" then I'd like to know so that I can be more aware of that in future.
I'm still unsure as to what I've done that was so heinous! I genuinely thought (and still do to some extent) that the two questions were duplicates, but decided that it was close enough to be worth asking on meta first, which I did. Since Tyler disagrees with me, I've accepted the fact of them being not close enough to be duplicates.
On the merits of the question itself, I don't think it's a great question. There's no evidence that the questioner has looked anywhere for any hints as to how to answer it, no evidence of a literature search or a google search - and it is a fairly obvious question to ask, and although searching exactly for L^p may not turn up anything (I don't know), to not think of varying L^p to a Banach or HIlbert space smacks of not thinking enough about the problem. That it led fedja to the question of uniformity is great, but (as I've said many times before) great answers do not make great questions. It's also gotten me pondering the subtleties of this issue since there's clearly something about uniformity that I didn't understand before, but that's still not enough for me to rate it highly as a question. However, I still think it's adequate for MO. That I don't approve of using MO as a first resort doesn't mean that I would go so far as to be actually rude to anyone who did so, nor that I haven't done it myself.
But that's a side issue to the question as to whether or not to "close as duplicate". Closing as duplicate is really a benefit for future visitors to the site to say, "If you think you're searching for this question, then actually you should be looking at this other question.". Since any answer to the one question is relevant to the other question, I do think that these two questions should be linked more closely than "If you liked this question, you might also be interested in ...". I agree that the answers to the second question could be answers to a different question that is not so closely linked to the first, but this question ain't that question. So "close as duplicate" is really just housekeeping for the site. In fact, if the phrasing had been "mark as duplicate", would anyone have objected?
However, let me repeat myself: I'm not going to vote to close as duplicate.
If I recall correctly, Andrew's original comment (now removed) said “technically, ‹other question› is a duplicate” (my emphasis). In my view, adding the word “technically” makes it clear that it is not obviously a duplicate, which should remove any (real or perceived) sting from the comment.
Andrew has a great suggestion (albeit one that we can't adopt yet): modify the user interface so that the process for "closing as duplicate" is separate from the usual closing process, and then change the wording indicating "closed as duplicate" questions.
Perhaps this could be as simple as removing "exact duplicate" from the list of reasons to close, and adding on option under "flag" for "mark as duplicate". Questions that receive enough flags as duplicates would then effectively be closed, but instead of marked as "[closed]" in the title, we could perhaps just write "[duplicate]".
If anyone is interested in considering this entirely theoretical proposal... Unfortunately even after the potential move to Stack Exchange 2.0, this would probably not be implemented by the Stack Exchange team, as I can't imagine them understanding our reasons for wanting this (i.e. that MathOverflow is extremely closure-averse relative to all the other Stack Exchange 2.0 sites).