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.