(1)

Let

(a) Find the limit points of .

(b) Is a closed set?

(c) Is an open set?

(d) Does contain any isolated points?

(e) Find

(For (e), use this definition. Given a set A , let L be the set of all limit points of A. Theclosureof A is defined to be = A 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 -neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set.

(a) -rational numbers

(b) -natural numbers

(c)

(d) (0,1]=

(e) {1+1/4+1/9+...+1/ }

(3) Let Prove that is an isolated point of A if and only if there exists an -neighborhood such that A= .

Any and all help would be greatly appreciated, thanks!