# Need Help with Cluster Points

• Mar 11th 2008, 01:33 PM
noname89
Need Help with Cluster Points
Can someone give me hints on how to solve this problem?
This is from Pugh's Real Mathematical Analysis
Problem 89.
Recall that p is a cluster point of S if each M_r_p contains infinitely many points of S. The set of cluster points of S is denoted as S'
Prove:
a) If S is a subset of T then S' is a subset of T'
b) (The union of S and T)'=Union of S' and T'
c) S' = (cl(S))'
d) S' is closed in M; that is, S" is a subset of S' where S"=(S')'
e) Calculate N', Q', R', (R\Q)', Q"
f) Let T be the set of points {1/n:n in N}. Calculate T' and T"
g) Give an example showing that S" can be a proper subset of S'
• Mar 11th 2008, 01:54 PM
Plato
If x is a point such that every neighborhood of x contains infinitely points of S then x is a cluster point of S and x is in S’.

If S is a subset of T and every neighborhood of x contains infinitely points of S, then does every neighborhood of x contains infinitely points of T? Does that prove that S’ is a subset of T’?

We know that $S \subseteq \left( {S \cup T} \right)$ does that mean $S' \subseteq \left( {S \cup T} \right)^\prime$? Now you can finish the second one.

All of the others have similar proofs.