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What is an accumulation point?

  1. Mar 25, 2013 #1
    I don't know what an accumulation point is and I have read the definition many many times.

    Could someone please give me a few examples with intervals of what would be an accumulation point?

    Thank you!
     
  2. jcsd
  3. Mar 25, 2013 #2

    SammyS

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    Hello Fire flame. Welcome to PF !

    What is the definition that you're trying to use, but don't understand?
     
  4. Mar 25, 2013 #3
    Definition:

    Let S be a subset of R. A point x in R is an accumulation point of S if every deleted neighborhood of x contains a point of S.

    So this is what I get out of it.

    Lets say S is a subset of R and S is the interval [0,1)

    So basically, you take a point, any point x in R and then deleted that point and look at the neighborhood around that point, if you get something in S in YES that point is an accumulation point.

    So from our interval [0,1)

    0 is an accumulation point since in the neighborhood to the right side of zero there is something in S
    1/2 or 0.5 is an accumulation point because on both sides of 0.5 there is a neighborhood that contains S
    1 is an accumulation point since to the left of 1 there is a neighborhood that contains S

    So is this the right way to thinking about accumulation points?

    Thanks.
     
  5. Mar 25, 2013 #4

    SammyS

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    That's a good start.

    For your example in which set S is the interval [0,1), every point in S is an accumulation point.

    Any point, a, to the left of 0 is not an accumulation point, because we can find a deleted neighborhood of a which doesn't intersect set S. such a deleted neighborhood is, (2a, a)∪(a, a/2) .

    1 is an accumulation point because every neighborhood immediately to the left of 1 intersects set S.
     
  6. Mar 25, 2013 #5

    Fredrik

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    Also interesting is that if x is an accumulation point of S, there's a sequence in S that converges to x. There's no "gap" between the accumulation point and the set; it's as close to the set as any point can get.
     
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