Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What does the term complementary interval mean?

  1. Sep 1, 2009 #1
    What does the term "complementary interval" mean?

    I'm studying the Cantor set and related notions right now, and I keep coming across the term "complementary interval." What does that mean? I can't find a definition spelled out anywhere!
     
  2. jcsd
  3. Sep 1, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: What does the term "complementary interval" mean?

    Where are you seeing this? How is it being used?

    Without knowing any context, I would guess that "complementary interval" is not a technical term -- it's simply an English phrase built from the two technical terms "complement" and "interval", or maybe even from the English word "complementary" and the technical term "interval".
     
  4. Sep 1, 2009 #3
    Re: What does the term "complementary interval" mean?

    Ok. Allow me to provide some context: "The difficult part is verifying continuity of the Cantor function at points of the Cantor set that have a sequence of complementary intervals converging to them."
     
  5. Sep 1, 2009 #4
    Re: What does the term "complementary interval" mean?

    I know nothing about what you are studying, but a wild guess: could it mean that the intervals don't overlap? If I and I' are any two intervals, then they are complimentary intervals if there exists no points in I which are in I' ?
     
  6. Sep 1, 2009 #5

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: What does the term "complementary interval" mean?

    The complement of the Cantor set in [0,1] is a countable union of disjoint open intervals. Maybe they're talking about those?
     
  7. Sep 1, 2009 #6
    Re: What does the term "complementary interval" mean?

    Yes, that's a reasonable idea. But how do you make sense of intervals NOT in the Cantor set converging to points that ARE in the Cantor set? What does it even MEAN for a sequence of open intervals to converge to a point?
     
  8. Sep 1, 2009 #7

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: What does the term "complementary interval" mean?

    Well, the most straightforward idea I can think of is:

    A sequence of sets Sn converges to a point P iff, for every open set U containing P, there exists an N such that n>N implies Sn is a subset of U.

    Or equivalently, every sequence sn of points chosen so that sn is an element of Sn converges to P.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: What does the term complementary interval mean?
  1. What does this mean? (Replies: 2)

  2. What does this mean? (Replies: 8)

  3. What does ye mean? (Replies: 6)

Loading...