What is an accumulation point?

[Fire flame]

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!

 

[SammyS]

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!

Hello Fire flame. Welcome to PF !

What is the definition that you're trying to use, but don't understand?

 

[Fire flame]

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.

 

[SammyS]

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.

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.

 

[Fredrik]

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.