Thread: Real Analysis - Open/Closed Sets

Decide whether 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 ε-neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set.

a. Q (the set of rationals)
b. N (the set of naturals)
c. {xЄR : x>0}
d. (0,1] = {xЄR : 0<x<=1}
e. {1 + (1/4) + (1/9) + ... + (1/(n^2)) : nЄN}

Some hints:
b) N is not open. Pick any x in N, for example 0. It's difficult to find an ε-neighborhood of x contained in N.

Note that the complement of N can be written as a union of open intervals.
(How?) This shows N is closed (Why?)

c) This is open but not closed. Look at 0 to see why it is not closed.

d) Looking at 1 will be helpful.