Accumulation Points: Set with Two or Countably Infinite?

• nitro
In summary, the conversation discusses the possibility of a set with a countably infinite number of accumulation points and a set with exactly two distinct accumulation points. An example of a set with two accumulation points is suggested, but the countably infinite case is left open for further exploration. Another example is proposed, involving a set in R that depends on two natural numbers and has a limit of zero, leading to a countably infinite number of accumulation points.
nitro

Homework Statement

is there are a set with countably infinite number of accumulation points?

Homework Equations

is there a set with exactly two distinct accumulation points?

The Attempt at a Solution

a set with two accumulation points might be: {1/n + (-1)^n}
i have no clue about the countably infinite one

hope someone could help!

thx a lot

nitro said:

Homework Statement

is there are a set with countably infinite number of accumulation points?

Homework Equations

is there a set with exactly two distinct accumulation points?

The Attempt at a Solution

a set with two accumulation points might be: {1/n + (-1)^n}
i have no clue about the countably infinite one
Your idea is fine, but be sure to write it correctly (with an ": n in N").

I'd prefer not to spoon feed you an example for the other; rather, consider perhaps a set (in R) that depends on two natural numbers. Say, $$\{y_n, y_n + x_m: n, m \in \mathbb{N}\}$$, which you might construct so that $$\lim_{m\to\infty} x_m = 0$$ and so, for each fixed n, $$\lim_{m\to\infty} y_n+x_m=y_n.$$

Last edited:
aahaaaaaaa! :)
thx a lot, i got it

What is an accumulation point?

An accumulation point, also known as a limit point, is a point in a set that can be approached arbitrarily closely by other points in the set. In other words, every neighborhood around an accumulation point contains infinitely many points from the set.

Can a set have more than one accumulation point?

Yes, a set can have multiple accumulation points. For example, the set of rational numbers has infinitely many accumulation points, including both rational and irrational numbers.

Is every limit point an accumulation point?

Yes, every limit point is also an accumulation point. However, the converse is not always true. A point can be an accumulation point without being a limit point if it is not in the set itself.

Can a finite set have accumulation points?

No, a finite set cannot have accumulation points. This is because a finite set only contains a finite number of points, so there will always be a neighborhood around each point that does not contain any other points from the set.

What is the difference between a set with two accumulation points and a set with countably infinite accumulation points?

The main difference is the cardinality of the set. A set with two accumulation points has a countable number of elements, while a set with countably infinite accumulation points has an uncountable number of elements. This means that the set with countably infinite accumulation points is larger and has more accumulation points than the set with only two.

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