Thread: Need Help with Cluster Points

1. 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'

2. 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.