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A question on intervals

  1. Jan 6, 2004 #1
    Please look at: http://mathworld.wolfram.com/Interval.html

    My question is about this line:

    "If one of the endpoints is +-oo , then the interval still contains all of its limit points, so (-oo,b] and [a,oo) are also closed intervals".

    How come ?
    Last edited: Jan 6, 2004
  2. jcsd
  3. Jan 6, 2004 #2
    If a sequence within [itex][a,\infty)[/itex] converges it must converge to a value within [itex][a,\infty)[/itex], and so the interval is by definition closed.

    Suppose you have a sequence that converges to [itex]p[/itex]. If [itex]p<a[/itex], then the sequence must contain a point in the interval [itex](p,a)[/itex] and so the sequence is not actually in the interval.

    So if the sequence is in the interval we must have [itex]p\geq a[/itex]. Thus [itex]p\in[a,\infty)[/itex]. Thus any convergent sequence in the interval converges to a point in the interval. Thus the interval is closed.
  4. Jan 6, 2004 #3


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    It depends on what definition of closed you're using, but [tex](-\infty,b][/tex] is indeed a closed set in [tex]\Re[/tex] under the usual topology.

    If you think of closed as 'contains all of its own limit points', then you can see that
    does indeed contain all real limit points of sequences of reals that only contain numbers from [tex](-\infty,b][/tex].

    This is relatively easy to prove:
    Let's say we have a sequence [tex]S[/tex] such that [tex]s \in S \rightarrow s \in (-\infty,b][/tex], and that [tex]x[/tex] is a limit point of [tex]S[/tex].
    Assume, by contradiction, that [tex]x[/tex] is not in [tex](-\infty,b][/tex]. Then clearly [tex]x > b[/tex]. Now, because [tex]x[/tex] is a limit point of [tex]S[/tex], for any [tex]\epsilon > 0[/tex] there exists some [tex]s' \in S[/tex] with [tex]|s'-x| < \epsilon[/tex], but if [tex]\epsilon=\frac{x-b}{2}[/tex] then there cannot be any suitable [tex]s' \in (-\infty,b][/tex]

    Alternatively, if you start with the notion that [tex](a,b)[/tex] is an open set in [tex]\Re[/tex] then you can see that [tex](b,\infty)[/tex] is an open set, since it is [tex]\Cup_{x \in \Re, x>=b} (x,x+1)[/tex]. Then it's compliment [tex](\infty,b][/tex] must be closed.

    P.S. Sorry, I don't have the tex for union handy.
  5. Jan 6, 2004 #4


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    If you are thinking that [itex](-\infty,b][/itex] and [itex][a,\infty)[/itex] do not include all limit points because they do not include [itex]-\infty[/itex] or [itex]\infty[/itex], remember that the those are not in standard real number system. That is why we never say "[itex][-\inft,b][/itex] or [itex][a,\infty][/itex].

    "[itex][a,\infty)[/itex]" really means "a and all real numbers larger than a".
  6. Jan 6, 2004 #5


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    If you are thinking that [itex](-\infty,b][/itex] and [itex][a,\infty)[/itex] do not include all limit points because they do not include [itex]-\infty[/itex] or [itex]\infty[/itex], remember that the those are not in the standard real number system. That is why we never say "[itex][-\infty,b][/itex] or [itex][a,\infty][/itex].

    "[itex][a,\infty)[/itex]" really means "a and all real numbers larger than a".
  7. Jan 7, 2004 #6
    When we find a 1-1 map between some point x to some R number, then if x in R then for any x in R, we can find some x0 < x OR some x < x2.

    Therefore x0 OR x2 are always unreachable for any given x.

    Let x0 be -oo(= inifinitely many objects < x).

    Let x2 be oo(= inifinitely many objects > x).

    No given x can reach x0 or x2.

    Therefore x0 OR x2 must be the unreachable limits of any R number.

    (x0,x] OR [x,x2), therefore [a,oo) OR (-oo,b] cannot be but half closed intervals.

    Therefore the set of all R numbers (where R has a form of infinitely many objects) does not exist.

    Shortly speaking, infinitely many objects cannot be related with the word all.

    Fore clearer picture please look at:

    Last edited: Jan 7, 2004
  8. Jan 7, 2004 #7
    i wouldn't say that oo or -oo in real analysis refers to infinitely many objects, at least no more than do numbers like 1 or square root of 2. actually, in the cauchy sequence construction and dedikind cut construction of R, all numbers are sets with infinitely many objects but of the same kind of infinity. neither oo nor -oo are real numbers, they aren't really defined except that oo is a symbol one could interpret as infinity if one wants such that x<oo for all x in R.

    quote from my real analysis book:
    the convienience is, for example, when talking about intervals like
    G={x in R : x>a}. now if F={x in R : a<x<b}, then the notation is (a,b). for G, this notation would become awkward: (a, (period)
    so to make the two notations look alike, we just say (a, oo) when we actually mean (a, (period) oo is not the right endpoint of G; there is no right endpoint of G.
  9. Jan 8, 2004 #8

    My argument is very simple:

    1) Things get infinitesimally small and never reach {}.

    2) Things get infinitely big and never reach {__}.

    Therefore things are in ({},{__}).

    {__} is the full set, which its content is an infinitely long non-factorized-one line.

    {__} is the opposite of {}, and vice versa.

    Please see again this example:


    Therefore no infinitesimally or infinitely many elements can be related to words like all or complete.

    Therefore definitions like 'the complete list of N numbers' are meaningless.

    The idea of transfinite universes is meaningless.
    Last edited: Jan 8, 2004
  10. Jan 8, 2004 #9
    Except that none of the things you're talking about have anything to do with math.
  11. Jan 8, 2004 #10
    Last edited: Jan 8, 2004
  12. Jan 8, 2004 #11
    i'm taking {__} to be for now the class of all sets or a universal object.

    1) a set with elements will never be the empty set

    2) a set not having all elements will never reach {__}.

    so, master_coda, can you explain how nothing has anything to do with math?
  13. Jan 8, 2004 #12


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    Well, the simple response is that the 'things' which Organic refers to are not mathematical objects.
    It follows that notions about big and small things are also not mathematical and so on.

    Hence, the argument that Organic makes is not mathematical in nature.

    For mathematical associations to be drawn -- for example the implied notion that there is a correlation between Euclidean (ie. Plane) geometry and boolean logic, and Non-Euclidean geometry and non-boolean logic -- they need to be formalized, or at least described.
  14. Jan 8, 2004 #13
    Well if all you have to do to talk about math is tie random math words together, then I would have to concede that what Organic says has something to do with math. Still, I would argue that he hasn't said anything meaningful or relevant.

    His argument seems to boil down to "you can't talk about infinite sets because you can't count to infinity". Not only is that wrong, but it also portrays a profound lack of understanding of the mathematical concept of infinity. Or even an understanding of logic, since he asserts that "for all" is not a valid quantifier.

    Organic asked a question about intervals. We posted a perfectly good answer to his question. Then we were told that our explaination was wrong. Apparently, when you change the definitions for everything, you get different results. Since we aren't told what those new definitions are, we can't even check these new results. It hardly matters anyway, since proving something about a different definition of "closed interval" doesn't prove anything about the original definition.
  15. Jan 8, 2004 #14


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    (For some reason I couldn't post this morning, but here's what I was gonna write)

    It seems you're confused about what x0 and x2 are supposed to be; when you first use them:

    They seem like they're supposed to be real numbers.

    However, when you next use them:

    They seem like they're supposed to be extended real numbers.


    Now x0 and x2 are sets!

    And now they're back to real numbers again (or maybe extended real numbers).

    Make up your mind, which is it?

    That doesn't prevent them from being closed sets. (and thus closed intervals)

    Heck, subsets of the real line can be both open and closed simultaneously! (in particular, [itex]\varnothing[/itex] and [/itex](-\infty, \infty)[/itex] have this property of being "clopen")

    This doesn't even have anything to do with the previous statements!! Why do you say this?

    In your theory, maybe, but it is not meaningless in ordinary mathematical structures.
  16. Jan 8, 2004 #15
    The contents of {} and {__} are total states that cannot be explored by any information system, including Math Language.

    Please read this: http://www.geocities.com/complementarytheory/MathLimits.pdf

    Therefore any information system is limited to ({},{__}) where:

    ({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

    Question: Is Universal set = {__} ?


    Universal set is
    the balance of ({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

    To understand this, Please read:





    The 'Boolean Logic' which is an Euoclidian-Mthematics system is goning to be replaced by systems like 'Complementary Logic' which is a Non-Euoclidian-Mathematics system.

    For example read this: http://www.math.rutgers.edu/~zeilberg/Opinion43.html

    Complementary Logic ( http://www.geocities.com/complementarytheory/CompLogic.pdf and http://www.geocities.com/complementarytheory/4BPM.pdf ) goes beyond the above article.

    Do you still do not realize that the Cantorian world is based on a private case of some broken symmetry?

    Please take a long look at:





    And if you understand the above, please take a long look at:





    And is you understand the above than please take a long look at:




    Last edited: Jan 8, 2004
  17. Jan 8, 2004 #16
    Hi Hurkyl,

    You are right, I don't no how to use standard notations, so to make my idea clear x0 and x2 stand for infinitely many R numbers where:

    x0 < some given x where x is any arbitrary R subset.

    x2 > some given x where x is any arbitrary R subset.

    The arbitrary R subset is represented by 01 infinitely long sequence,
    taken from set Rseq, which its content = [...000,...111)XOR(...000,...111]

    Rseq cunstructed by:
    Therefore it can't be "clopen".

    About "clopen" I have found this:

    Can you please explain "clopen" in a non-formal way?
    Last edited: Jan 8, 2004
  18. Jan 8, 2004 #17
    it is much more useful to actually point out the errors than to just say they are there. what's going on here is two differnet notions of intervals, which sort of resemble each other but not in a "rigorous" way, are being mixed up. ({}, {__}) has nothing to do with intervals like (a,b) for the first one, for one thing, is about much more than real numbers. the way i'd put it is that there is a "lattice" with {} at the bottom and {__} at the top. however, since not everything in between is "comparable," it doesn't make sense to use interval notation, in which there usually is a "total ordering," or at least a "linear ordering," involved in all elements in the interval.

    that is, i think, a straw man. that can't be his argument because he's talking about infinite sets. i also think he's referring to the absolute infinity and not just any infinite set. and in that sense, you can not count to the absolute infinity no matter what. that is to say i can prove, i think, that if P(X) is the absolute infinity, then X is the absolute infinity; hence, it cannot be achieved "from below." iow, you cannot count to it or power set to it. i think if you just unravel what organic is trying to say and get past the fact that he's using nonrigorous language, he's got kernels of truth.

    in fact, he was using these statements on infinite sets like N, you know that N can be approached but never achieved. he had the right idea but what he means, i think, is the universal object, not N. when he did, i argued up until when i realized what he was really talking about.

    i agree that if you change the definitions mid-sentence or mid-article, you have huge problems and that's something he has to work on but i think his nuggets of truths should be encouraged and we, like hurkyl, should be correcting the language rather than simply say it is incorrect. if that's not worth your time, i understand but if you just say it's incorrect without correction, that's not really worth organic's time.

    ps: organic, they did try to correct your language but you didn't seem to listen! you have to make it clear that you're not talking about the same kind of intervals. you have to define what you mean and stick to it. look at any definition on mathworld.com or any textbook and make your definitions look like that. believe me, it's not so limiting to stick to that.
    Last edited: Jan 8, 2004
  19. Jan 8, 2004 #18
    Hi phoenixthoth,

    Please read my previous post (my answer to Hurkyl), and reply your remarks.

    Thank you,

    Last edited: Jan 8, 2004
  20. Jan 8, 2004 #19


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    Literally, "clopen" means "open and closed".

    In nice situations, it turns out there aren't very many clopen sets; in some of the more common topological spaces, like [itex]\mathbb{R}[/itex] or [itex]\mathbb{C}^n[/itex], the only two clopen sets are the empty set and the entire space; in nice situations the only clopen sets are those that are unions of the connected components of a space; that is, you pick a few (maybe zero) points, and you get a clopen set by taking all of the points that can be connected to the selected points.

    So, for example, if I make a topological space by choosing three disjoint lines l, m, and n, then the clopen sets are [itex]\varnothing[/itex], l, m, n, l U m, l U n, m U n, l U m U n.

    In not so nice situations (e.g. the rational numbers), there can be more clopen sets. For instance, the interval [itex](\sqrt{2}, \sqrt{3})[/itex] is clopen in the rational numbers; it's closed because it contains all of its limit points in the rational numbers, and it's open because it has no boundary points in the rational numbers.
  21. Jan 8, 2004 #20
    and a space could be called "connected" if and only if the only clopen sets are Ø and the whole space. so when hurkyl says nice, could mean "connected."
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