Cluster points

[passionate]

Would anyone show me how to find sets of cluster points?
For example how to find the sets of cluster points of N, of R, of R\Q, of Q'. Where Q' denotes the set of cluster points of Q.

 

[pka]

Would anyone show me how to find sets of cluster points?

Please define cluster points.
If you post, please explain the terms.
Form what you post I would guess that these are just limit points from topology.
However,” cluster points" are used in lattice theory.
We don’t know what you are studying.

 

[passionate]

Sorry for the confusion.
p is a cluster point of S if each M_r p contains infinitely many points of S. From what I learned, it's a type of limit point.

 

[pka]

This is a rather odd question given the simplicity of the answers.
The integers have no cluster points.

The rest of the question is trivial, here is why. Every real number is the limit point of a sequence of distinct rational numbers.
Thus very real number, rational or irrational, is a cluster point for the set of irrationals and of the set of rationals.

If \(\displaystyle H'\) stands for the set of cluster points of the set \(\displaystyle H\) then.
\(\displaystyle \Re =\Re'\), \(\displaystyle \Re =Q'\), \(\displaystyle \Re=\left( {\Re \backslash Q} \right)^\prime\).

 

[stapel]

p is a cluster point of S if each M_r p contains infinitely many points of S.

Just to clarify: Would it be correct to say that "M[sub:hwh8eeh7]r[/sub:hwh8eeh7](p)" is a neighborhood of p (an open set, maybe), perhaps centered at p, of radius r...?

Thank you!

Eliz.

 

[passionate]

This is a rather odd question given the simplicity of the answers.
The integers have no cluster points.

The rest of the question is trivial, here is why. Every real number is the limit point of a sequence of distinct rational numbers.
Thus very real number, rational or irrational, is a cluster point for the set of irrationals and of the set of rationals.

If \(\displaystyle H'\) stands for the set of cluster points of the set \(\displaystyle H\) then.
\(\displaystyle \Re =\Re'\), \(\displaystyle \Re =Q'\), \(\displaystyle \Re=\left( {\Re \backslash Q} \right)^\prime\).

Thanks a lot for your help. I understand the definition better now. So based on what you showed me. I found that N' is empty, and 0' is empty too.
Let M= {1/n : n is natural number}, then I find that M' = 0. At first, I didn't really get the definition of a cluster point. So, now I know that informally it's a point that we can choose a small interval wraps around it, then that small interval must have infinitely many points belonging to the given set. For instance, if the set is Q, then picking any point in Q and no matter how small the radius, there are infinitely many rationals in that interval due to the density of Q. Again, thank you very much for the clarification.