What funny books you read! - But more seriously: In my copy of Rudin's "Real & Complex Analysis" I read the following: "If , theinterval[a,b] and thesegement(a,b) are defined to be " (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 ."

Note that here, Rudin does not even bother to require that (and much less that ) 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.

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

This is odd, don't you think?

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 un ensemble ordonné, et deux éléments de E tels que ." So, as expected, I can quote the official "Bourbaki" on my side as well.