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  1.  
    • CommentAuthorIan Durham
    • CommentTimeFeb 13th 2010
     

    Now that, hopefully, everyone has calmed down a bit, I have made some changes to my problematic question (http://mathoverflow.net/questions/14162/finite-versus-infinite-on-non-hausdorff-topologies-closed) and am wondering if people could take a look at it to see if it now makes more sense (and satisfies MO criteria for a reasonable question).

    I should like to add that I'm surprised a bit at the reactions I got over my definition of a Hausdorff topology. It seems odd that no one caught the fact that it was simply a typo on my part (substituting $\ne$ for = by accident), though this assumes that my current definition is correct.

    Thank you.

    • CommentAuthorMariano
    • CommentTimeFeb 13th 2010
     

    The question surely makes sense. I do not think it is very appropriate for MO, since it is asking for an example that should have come up in a general topology course so it is not quite a "research level math question".

    I think that the definition of the Hausdorff property is really not needed---as essentially everyone on the site should know it!---and that the example is an extraordinarily elaborate example, involving concepts several orders of magnitude more complicated, than what it is (as far as I can see) supposed to exemplify.

  3.  
    A related question:

    http://mathoverflow.net/questions/9267/what-is-an-explicit-example-of-a-sequence-converging-to-two-different-points

    You should be able to adapt answers there to get what you want.
    • CommentAuthorIan Durham
    • CommentTimeFeb 13th 2010 edited
     

    Well, I will admit to being entirely self-taught in the area of topology and thus am not as familiar with examples of the sort I am seeking and I'll be perfectly content if someone here on meta simply gives me an example or a reference.

    Edit: Which it appears someone did just as I posted this reply. Muchas gracias.

  5.  
    @Dr. Durham: "It seems odd that no one caught the fact that it was simply a typo on my part."

    ?!?!

    From the comments:

    " @Ian: There must be a typo in that book if that is verbatim, because not only is that not the definition of Hausdorff, but it's an entirely useless definition! Hausdorff replaces to $\ne$ with $=$. The definition yuo give is satisfied by every topological space: take the open set ot be the whole space. So this can't just be an example of disconnect, if people are seriously working with that definition, they're just doing point-set topology with no Hausdorff condition. – Charles Siegel Feb 5"

    " @Ian: The only explanation then is that it's a typo (if your last "$\neq$" were a "=", it'd be correct) - not some kind of linguistic disconnect.... – Zev Chonoles Feb 5"

    The revisions you made on your question this morning make it much clearer. There is still some ambiguity, since Zev Chonoles and I interpreted it in two different ways (the answers are similar, though). If, after looking at these comments, you want to post a new version of the question, I guarantee you will get answers within the hour. However, I predict that you will also get some comments along the lines that your question is a very basic one in undergraduate topology and not really appropriate for a forum for research mathematicians. If that response is going to distress you, then it is probably safer just to think through the comments on your existing question, which already provide reasonable answers.
    • CommentAuthorIan Durham
    • CommentTimeFeb 13th 2010
     

    Well, you've got me there. Apparently Zev did notice it was a typo and I seemed to miss that. Nevertheless it seemed like a rather big "to-do" when someone could simply have changed the typo.

    That said, I will note that there are questions on here that are basic undergraduate probability questions but no one seems to knock them. There seems to be an assumption that all the folks on this site have the same sort of background.

  7.  
    In fact, Pete noticed the typo first:

    "the current version defines a Hausdorff space -- which seems unnecessary, BTW -- but the definition given is incorrect"

    and you responded that this is what you thought was indicative of "a serious disconnect."

    As to why it wasn't changed, I suspect: 1) no one following the question had both the edit capabilities and the motivation to change it, since those people realized that 2) everyone else was ignoring it, giving you the benefit of the doubt that you knew the correct definition, until 3) you defended the incorrect definition, at which point (I assume) leaving the original was important so that people could easily refer to it.
    • CommentAuthorIan Durham
    • CommentTimeFeb 13th 2010
     

    I guess I didn't read Pete's comments as saying he thought it was a typo. I thought I was defending the correct definition (which I obviously wasn't - sometimes no matter how many times one looks at something, if they're tired enough they won't see the error right in front of their face).

  9.  
    @Dr. Durham: We all expected you to change the typo. Although it is possible for sufficiently high rep users to edit questions, most of us are hesitant to do so (unless the question is CW). We respected the possibility that you meant what you said enough not to change it. Rather, we tried (repeatedly) to call your attention to it in the comments.

    Here are some examples of basic (undergraduate and graduate) level probability questions that have been "knocked":

    http://mathoverflow.net/questions/14989/coupling-of-two-non-independent-random-variables-closed
    http://mathoverflow.net/questions/14867/uniform-distribution-probability-x-outside-2-standard-deviations-0-closed
    http://mathoverflow.net/questions/14762/exploading-levy-processes
    http://mathoverflow.net/questions/14120/problem-with-pearson-correlation-coefficient-closed
    http://mathoverflow.net/questions/10732/probability-of-coin-flipping-and-independent-events-closed
    http://mathoverflow.net/questions/12462/limsup-and-liminf-for-a-sequence-of-sets-closed
    http://mathoverflow.net/questions/7949/a-probabilistic-inequality-closed

    As for assumed background: yes, this is a site for graduate students and research mathematicians, and for those who can participate in that level of discourse. I would (and did, up until now) assume that a research mathematician (pure or applied) has had undergraduate topology. I am on the graduate committee of the math department (which contains both pure and applied mathematicians) of UGA -- we are something like the 50th best department in the US -- and the vast majority of students we admit to our PhD program have had a topology course. In order to get a PhD, our students must pass an exam in basic graduate topology.

    I cannot assume that a PhD in mathematics has had a course in probability, because this assumption is demonstrably false for a substantial percentage of math PhD's. This is a pity -- many pure mathematicians are ignorant of the simplest probabilistic ideas, to their detriment. I have given a "remedial" talk on probabilistic methods in analytic number theory:

    http://math.uga.edu/~pete/probnumtalk1.pdf

    But it is the stated purpose of MO to entertain questions which are above the undergraduate / basic graduate textbook level in any area. If you post a question on MO, do your homework first: check whether it is answered in a standard textbook on the subject. Check whether it is addressed on wikipedia. Think about it yourself for a while. Or don't do these things and take your chances (I have asked at least one question here that turned out to be on "textbook" commutative algebra), but be prepared to get "knocked" a little bit by the experts. Getting knocked isn't fatal; it might even be beneficial.
  10.  
    @Zev: yes, I called attention to the typo in my closing comment. I had actually quoted myself in my first comment in this thread above, and mentioned that "proofread"ing is what you do to fix typos, not serious mathematical mistakes. I honestly thought that what I wrote would be sufficient for Dr. Durham to easily recognize and correct his error. Obviously it wasn't. So to simplify the discussion I went back and deleted this part of my response.
  11.  
    @Pete: Ah, ok. Also, your description of MO in the last paragraph is really spot on. For example, I certainly deserved to get knocked on the converse to Hilbert's basis theorem.
    • CommentAuthorIan Durham
    • CommentTimeFeb 13th 2010
     

    @Herr Dr. Pete: note that it is actually possible to obtain a BS in Mathematics from MIT without taking a formal course in topology: http://www-math.mit.edu/academics/undergrad/major/. Nevertheless, I will make sure I mention your concerns to the head of our math department since we don't require a full course in it (and before anyone says anything, I should note our math and physics graduates are presently at places like Yale, Dartmouth, Tufts, Boston College, etc.).

    I am very pleased that you think there is a serious deficit in knowledge of probability theory among many mathematicians. It is a pet peeve of one of my colleagues in our math department and we were just talking about it yesterday.

    In any case, while not an example from probability, here is a question that I thought was at an undergraduate level (since I taught this question and the given answer in a class a few years ago) that didn't get knocked down. I didn't knock it down because I didn't assume the person asking had had a background in the material:

    http://mathoverflow.net/questions/14980/hat-problem-hamming-codes

  13.  

    The mathematics major at MIT has a few subdivisions, and the theoretical option does in fact require our topology course. I should add that one big reason not to require the topology course of non-theoretical math majors is that here a math / computer science double is much more common than a math / physics double, which is why the applied option has a strong emphasis on discrete math. (Also, the topology course is regarded as difficult - it's taught about half the time by Munkres, who has a very take-no-prisoners attitude about the whole thing.)

    As far as the Hamming code problem, you'll note that it only has one vote, so it isn't exactly a great question. Probably the OP should have thought about the problem a little more.

  14.  

    @Qiaochu: I think we should be careful in assuming that few votes implies not-greatness. It could just as well have few votes because few people on MO are interested (or are qualified to judge). This is not a comment on the Hamming code problem; I am making a general statement here. (I am not the one who voted it up, though.)

    • CommentAuthorIan Durham
    • CommentTimeFeb 13th 2010 edited
     

    And apparently I'm not the only one who thinks there is room for debate on just how research-oriented a question is: http://tea.mathoverflow.net/discussion/216/stop-closing-probability-questions-if-you-dont-know-probability/

    @QY: I never said the Hamming code problem was good. It was simply an example of a problem that I considered undergraduate-level that did not get closed and/or knocked into the negative numbers.

    But we digress. This thread was to be about my question on Hausdorff topologies and how I could improve it. I will be happy to debate undergraduate education and what makes a good research-level question on another thread.

    • CommentAuthorHarry Gindi
    • CommentTimeFeb 13th 2010 edited
     
    I suggest you read one of the following three books:
    Munkres - Topology
    Kelley - General Topology
    Bourbaki - General Topology I.

    If you do, this, your answer will be clear. If you don't know topology, then now is the time to learn. It is not hard. I have faith in you.
  17.  
    @Dr. Durham,

    I think you are missing some nuances of what I'm saying. (At least I hope you are missing them, and not just ignoring them for rhetorical reasons.)

    I didn't say that I assumed that anyone with a bachelor's degree in mathematics had taken an undergraduate topology class. I said that I assumed that anyone with a PhD in mathematics had an undergraduate topology class [or equivalent knowledge]. At UGA our undergraduate topology class is certainly not required of all math majors, and the percentage of all students who graduate with a BA/BS in math from UGA and have taken undergraduate topology is less than 50%. The ones who take it are mostly intending to go on to grad school, and conversely.

    It does follow from this that students applying to, say, UGA from a school which does not even offer a topology course will be at a competitive disadvantage and, if admitted, will have some catching up to do. There's nothing wrong at all with catching up: usually our undergraduate topology class has at least one graduate student.

    Secondly, I didn't say that you or any particular person should knock down everyone whom you think has asked a too elementary question. That's a judgment call every time. Some of the most eminent mathematicians who participate on MO remain above the fray and never say things like this. (I do agree though that sometimes they should be said.)

    @Harry: thanks for pointing out some basic texts. The title of Kelley's book is, indeed, _General Topology_. Stop me if you've heard this before -- I don't think Bourbaki's book is a good choice for someone who has not had an undergraduate course.

    In my opinion it should be possible nowadays to learn basic general topology without even having to take the trouble to go to the library (a sufficiently mathematical library may not exist conveniently close to you) or to buy one (you may not have the money to spare). That is, there should be resources available on the internet. And there are: e.g.

    http://www.math.ku.dk/~moller/e03/3gt/notes/gtnotes.pdf

    Unfortunately none of the free online texts I've seen really compete with Munkres or Kelley. There is room for improvement here...
  18.  

    Ian, I didn't like your phrase "Herr Dr Pete". I assumed the idea was something like: "Herr Dr" is German, German means Nazi, the Nazis told people what to do, and Pete was telling you what to do. Step number two is pretty offensive.

    Tell me I jumped to the wrong conclusion, and that will make me happier.

    • CommentAuthorYemon Choi
    • CommentTimeFeb 13th 2010
     

    @Tom: my charitable reading was that it was a jibe at credential-waving, using the fact that the German conventions keep all titles cumulatively (IIRC), thus "Herr Doktor Professor Weierstrass", which to an anglophone ear sounds excessively stilted. Like Thaddeus Grommit, BA, MA, MD, etc. It doesn't read so badly, I think, in the context of the discussion between Ian and Pete on what is and isn't covered in graduate training; although I didn't particularly care for it as rhetoric, I didn't think it was a Godwin.

  20.  

    (I don't think the Germans accumulate titles like that. But “professor” and “doctor” are somewhat orthogonal concepts, with neither implying the other, so it kind of makes sense (if you care about such things) to include both. They don't pile on their undergraduate degrees, though. This entire comment is in parenthesis in order to be more easily ignored.)

    • CommentAuthorKevin Lin
    • CommentTimeFeb 13th 2010
     

    ( http://www.google.com/search?q=%22Herr+doktor+professor%22 )

  22.  
    I was not offended by Dr. Durham's Germanic allusion.

    I call Dr. Durham "Dr. Durham" because he has informed me that he found some of my previous remarks disrespectful. When referring to someone in a professional context, especially if one is making remarks which are at all critical, unless a certain friendly familiarity is presumed to exist, Dr. So-and-so seems appropriate. Some people have gone so far as to suggest that Dr. Durham is, somehow, "not really a PhD". This is of course ridiculous, so it does not hurt to remind people that he is.

    It is a mark of the friendly respect that we (usually!) show each other here on MO that people of very different levels feel free to "tutoyer" each other. I rather enjoy it, although I should say that just because in mathematics people don't stand much on ceremony, it doesn't mean that there is no community sentiment that we should not go out of our way to be respectful to our elders and betters: most people do indeed feel this way. (I should add that I also feel that we should go out of our way to be respectful to our peers and also to those that are younger / less advanced in their studies or work. It's only that the way to show respect to these different groups is not always exactly the same. Thus at first I thought it was weird that the user page asks for your age, and I didn't fill it in immediately, but I see now that this can be helpful.)
  23.  

    Pete, I wasn't really expecting you to be offended, because I assumed you would have said so if you had been. People trying to be funny sometimes blur the concepts of German and Nazi. (This is particularly common in Britain, unfortunately.) If I were German I really wouldn't like this, but it's generally objectionable to everyone, really.

    Anyway, Yemon's charitable reading may be right. Looking back at the various comments, it seems very plausible, and it's certainly a happier interpretation.

    • CommentAuthorIan Durham
    • CommentTimeFeb 13th 2010
     

    @Tom: Actually I don't really think I meant it either way (though my meaning was closer to Yemon's). It was a Mel Brooks-ism and was supposed to inject a little light-heartedness into things. I have never heard it interpreted in the way in which you are interpreting it and I'm middle-aged. The Nazis were atrocious butchers but not all Germans were Nazis (my wife's family, for instance, is German Jewish). In addition, that title is not exclusive to Germany and Austria (Switzerland and Liechtenstein both employ it I believe).

    @Pete: I appreciate the respect, but you can return to calling me Ian. Whether you meant it or not, after our blow out, it comes off as patronizing.

  25.  

    Ian: OK, thanks for clarifying. I've heard too much of this type of "humour", which as I said is sadly prevalent here in the UK, probably more than elsewhere. Sorry to have been unnecessarily suspicious.

    • CommentAuthorHarry Gindi
    • CommentTimeFeb 14th 2010 edited
     
    @Tom: Ian is originally from the UK though.
    • CommentAuthorIan Durham
    • CommentTimeFeb 14th 2010
     

    @fpqc: I got my PhD in Scotland but I was (very proudly) born and raised in Buffalo, New York.

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