Give an example of A subset S of R which is neither open nor

Give an example of

A subset S of R which is neither open nor closed and every real number is a cluster point of S. S =

Solution

Solution :

Let S = (2, 6) and let T = [1, 5]. Then S is open and T is closed. Furthermore, S T = (2, 5]. That set is not closed because 2 is an accumulation point and not in the set. The set (2, 5] is also not open because (2, 5] contains 5, but there is no open interval (a, b) containing 5 such that (a, b) (2, 5].