Finding cluster points of a set?

[cookiesyum]

Homework Statement

What are the cluster points for the set

S = {all (1/n, 1/m) with n = 1, 2, ..., m = 1, 2, ...}

Homework Equations

A point p is a cluster point for a set S if every neighborhood about p contains infinitely many points of the set S.

The Attempt at a Solution

The graph of the set is the open square formed by the points (0,0) (0,1) (1,0) (1,1)?

The book says cluster points include those such as (0, 1/n) for each n and others on the horizontal axis, as well as the origin. I don't really understand why though...

 

[Dick]

No, the graph isn't the open square. It's discrete points in the open square. All of the cluster points are on the boundary of the open square. Do you understand why (1,0) is a cluster point?

 

[lurflurf]

Think about when neighborhoods of points in your set pile up.

 

[cookiesyum]

No, the graph isn't the open square. It's discrete points in the open square. All of the cluster points are on the boundary of the open square. Do you understand why (1,0) is a cluster point?

The points become closer together near the vertical and horizontal axes. I think that's why the cluster points are on the vertical and horizontal axes. Or, they are of the form (0, 1/m) (1/n, 0) and (0, 0). But why is there a cluster point at, say, (1, 1)?

 

[Dick]

There isn't a cluster point at (1,1). The nearest point to (1,1) is (1/2,1/2), isn't it? Hardly a cluster point.

 

[cookiesyum]

There isn't a cluster point at (1,1). The nearest point to (1,1) is (1/2,1/2), isn't it? Hardly a cluster point.

Ok, thank you so much!