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Why is x > 0 not an open interval?

  1. Mar 19, 2010 #1

    This might be a trivial question, but I'll ask it anyway.

    Why is this not an open interval? Let a, b be two elements in X, such that a < b. Then, every real number c satisfying the order relation a < c < b is necessarily in (a, b) and hence in X. Hence X is an interval. Since X is equivalent to the interval (0, [itex]\infty[/itex]), I assumed it is also an open interval. Does the definition of an open interval only apply to intervals with finite end points? I thought otherwise: http://en.wikipedia.org/wiki/Interval_(mathematics)#Infinite_endpoints.

    Thanks in advance.
  2. jcsd
  3. Mar 19, 2010 #2
    I don't know the reference and thus the context, but are you sure the authors are not referring to a 'bounded interval' ?

    Or are the authors distinguishing between the reals (which do not contain [tex] \pm \infty [/tex] ) and the extended reals which do?
    This distinction is often ignored.
  4. Mar 19, 2010 #3
    Look in that book for the definition of "open interval". Your set is an open set and an interval, but perhaps there is a technical definition of "open interval" used in that book, requiring an "open interval" to have the form (a,b) with b a real number.
  5. Mar 19, 2010 #4
    Thanks for the replies. This is what the book says

    No mention has been made of whether a and b must necessarily be finite.
  6. Mar 19, 2010 #5
    and what is the book's definition of R ?

    does it contain infinity (the extended reals)

    or not (the reals)
  7. Mar 19, 2010 #6
    What are a and b? Most likely, the book wants a,b to be elements of R.
  8. Mar 19, 2010 #7
    Nothing else is listed explicitly. I would assume that a and b are elements of R too. But no mention is made of whether a and b can be infinite.
  9. Mar 19, 2010 #8
    If it turns out out that a,b are supposed to be members of R, then it makes sense that (5, infinity) is not an open interval (because open sets are the ones of the form (a,b) where both a AND b are in R. (infinity is not in R).
  10. Mar 19, 2010 #9
    Ah. Thanks.
  11. Mar 25, 2010 #10
    I guess you meant to write "open intervals" instead of "open sets". (The example was said to be an open set but not an open interval.)
  12. Mar 25, 2010 #11
    You read my mind, thank you for the correction.
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