In my real analysis text, I have read, reread, re-re-read, and re-re-re-read the definition of an accumulation point. Then after letting my thoughts marinate for awhile, I read it again.
Definition from text: "A point p is an accumulation point of a set S if it is the limit of a sequence of points of S - {p}. It is equivalent to require that every ball (p - r, p + r) about p intersect S - {p}."
One of my fellow classmates gave me the following advice: 'stand on a point and stretch out your arms. What do you hit?' I think that was the missing puzzle piece I needed to put together the following:
1. A "ball" is just a range of points. In 1-D space, a ball would be an interval; in 2-D space, a ball would be a circle, etc... So then r would be the 'radius' of my interval or circle or sphere or ....... depending on the space I'm in.
2. If I have a set of points, and I'm standing on a point somewhere, I should be able to create any size ball and include at least one point in the original set, NOT INCLUDING the point I'm standing on.
3. Example: I'm standing on a number line, my set S is the rational numbers, and I stand on the point 4. I can create an interval (ball) that hits at least one rational number, therefore 4 is an accumulation point. I can also stand on the point 4.1564654 and do the same thing, therefore 4.1564654 is an accumulation point, as is every single real number. In fact, I can ALWAYS hit a rational with any size interval/ball by standing on ANY real number. Therefore all the Real Numbers are accumulation points.
4. Another example: I'm still standing on a number line, my set S is the integers, and I stand on the point 4. I create an interval that is 0.13 on either side of 4, hitting no other integer. I cannot include 4, because I'm standing on it. Therefore 4 is not an accumulation point. In fact, there are no accumulation points, because I MAY hit an integer with my interval, but I cannot GUARANTEE I will hit an integer.
Am I understanding accumulation points correctly? Thanks in advance for any assistance.
Definition from text: "A point p is an accumulation point of a set S if it is the limit of a sequence of points of S - {p}. It is equivalent to require that every ball (p - r, p + r) about p intersect S - {p}."
One of my fellow classmates gave me the following advice: 'stand on a point and stretch out your arms. What do you hit?' I think that was the missing puzzle piece I needed to put together the following:
1. A "ball" is just a range of points. In 1-D space, a ball would be an interval; in 2-D space, a ball would be a circle, etc... So then r would be the 'radius' of my interval or circle or sphere or ....... depending on the space I'm in.
2. If I have a set of points, and I'm standing on a point somewhere, I should be able to create any size ball and include at least one point in the original set, NOT INCLUDING the point I'm standing on.
3. Example: I'm standing on a number line, my set S is the rational numbers, and I stand on the point 4. I can create an interval (ball) that hits at least one rational number, therefore 4 is an accumulation point. I can also stand on the point 4.1564654 and do the same thing, therefore 4.1564654 is an accumulation point, as is every single real number. In fact, I can ALWAYS hit a rational with any size interval/ball by standing on ANY real number. Therefore all the Real Numbers are accumulation points.
4. Another example: I'm still standing on a number line, my set S is the integers, and I stand on the point 4. I create an interval that is 0.13 on either side of 4, hitting no other integer. I cannot include 4, because I'm standing on it. Therefore 4 is not an accumulation point. In fact, there are no accumulation points, because I MAY hit an integer with my interval, but I cannot GUARANTEE I will hit an integer.
Am I understanding accumulation points correctly? Thanks in advance for any assistance.