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Math Help - What is NOT an interval?

  1. #16
    Super Member Failure's Avatar
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    Quote Originally Posted by Plato View Post
    How very odd. Your edditions/translations of Rudin and Dieudonne differ so from mine.
    In both 'Big Rudin' and 'Little Rudin' that I have, this definition appears:
    “The segment (a,b) is the set of all real numbers, x , such that a<x<b.”
    Then he notes that a interval is a segment that includes its endpoints.
    To me that rules out {a} being an interval.
    What funny books you read! - But more seriously: In my copy of Rudin's "Real & Complex Analysis" I read the following: "If -\infty \leq a\leq b\leq \infty, the interval [a,b] and the segement (a,b) are defined to be [a,b] = \{x: a\leq x\leq b\}, \quad (a,b) = \{x:a<x<b\}" (1.1., Page 7 of the 1981 reprint)
    So, Rudin here does not really call (a,b) an interval: he calls it a segement. But I think this is a side issue.

    In "Principles of Mathematical Analysis" I read, on page 31, 2.17 Definition: "By the interval [a,b] we mean the set of all real numbers x such that a\leq x\leq b."
    Note that here, Rudin does not even bother to require that a\leq b (and much less that a<b) holds. If no x exists between a and b, the interval in question is just the empty set, which he also takes to be an interval.

    Virtually the same statement appears on page 17 of my translation of Dieudonne.

    This is odd, don't you think?
    As regards Dieudonné, it is true that his phrasing in my German translation of "Foundations of Modern Analysis" is a little bit confusing. But he makes it absolutely clear by adding that in the case of a=b, the notation [a,b], i.e. [a,a] means {a}, as tonio has already mentioned.
    As I see it, Rudin just has found a more succinct way of getting his meaning across in all cases, while Dieudonné here fumbles a bit (in my humble opinion) but finally gets around to making things clear as regards our bone of contention.

    Edit: Let me add that Bourbaki, "Éléments de Mathematique: Théorie des Ensembles" in E III.14, starts out defining intervals with the sentence: "Soient E un ensemble ordonné, a et b deux éléments de E tels que a{\color{red}\leq} b." So, as expected, I can quote the official "Bourbaki" on my side as well.
    Last edited by Failure; October 30th 2009 at 11:03 PM.
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  2. #17
    Super Member Failure's Avatar
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    Quote Originally Posted by Plato View Post
    How very odd. Your edditions/translations of Rudin and Dieudonne differ so from mine.
    In both 'Big Rudin' and 'Little Rudin' that I have, this definition appears:
    “The segment (a,b) is the set of all real numbers, x , such that a<x<b.”
    Then he notes that a interval is a segment that includes its endpoints.
    To me that rules out {a} being an interval.
    Ah! Now I see what your problem is: it all depends on whether the author requires that a<b or whether he is satisfied with the weaker condition that a\leq b or even with no condition on a and b at all (like Rudin in his "Real & Complex Analysis").
    If an author, such as Rudin in "Principles of Mathemacal Analysis", requires only -\infty\leq a{\color{red}\leq} b\leq \infty, it follows that an interval like [a,a] is equal to the set \{x\in\mathbb{R}: a\leq x\leq a\} which is clearly the same as \{a\}. And the segment (a,a) would simply be the empty set.

    Since Rudin is satisfied with the weaker condition a\leq b, he follows the usage that I suggested was the more common. So it seems to me that your reading of Rudin's text is wrong.
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