Open and Closed Sets

**Proof_of_life** · May 27th 2008, 11:12 AM

(1)
Let $B= \{\frac{(-1)^{n}n}{n+1}: n =1,2,3,...\}$

(a) Find the limit points of $B$ .
(b) Is $B$ a closed set?
(c) Is $B$ an open set?
(d) Does $B$ contain any isolated points?
(e) Find $\overline{B}$

(For (e), use this definition. Given a set A $\subseteq R$ , let L be the set of all limit points of A. The closure of A is defined to be $\overline{A}$ = A $\cup$ L

(2) Decide wheather the following sets are open, closed, or neither. If a set is not open, find a point in the set for which there is no $\epsilon$ -neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set.

(a) $Q$ -rational numbers
(b) $N$ -natural numbers
(c) $\{x\in R: x>0\}$
(d) (0,1]= $\{x\in R: 0<x\underline{<}1\}$
(e) {1+1/4+1/9+...+1/ $n^2: n \in N$ }

(3) Let $a \in A.$ Prove that $a$ is an isolated point of A if and only if there exists an $\epsilon$ -neighborhood $V_{\epsilon}(a)$ such that $V_{\epsilon}(a) \cap$ A= ${a}$ .

Any and all help would be greatly appreciated, thanks!

**Proof_of_life** · May 27th 2008, 02:13 PM

Sorry, is this the right forum for this type of question? This is from my analysis class. Thanks.

**Proof_of_life** · May 28th 2008, 09:29 AM

I have solved (1) and (2), so if anyone could take a crack at #3 that would be coolio!

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