Let's analyze each statement:
1) S is an open set.
- False. S is not open because it does not contain all points in some neighborhood of each of its points. For example, 1 is in S but there is no open ball around 1 that is fully contained in S.
2) S does not have any accumulation point.
- False. 1 is an accumulation point of S.
3) i is an accumulation point of S.
- True. Every open ball centered at i contains points of S.
4) T is an open set.
- False. T is not open because it does not contain all points in a neighborhood of each of its points. For example, 1 is in T but no open ball around 1 is fully contained in T.
5) T is a closed set.
- True. T contains all its boundary points.
6) i is an accumulation point of T.
- True. Every open ball centered at i contains points of T.
7) 1 is an accumulation point of set T.
- True. Every open ball centered at 1 contains points of T.
8) S is neither open nor closed set.
- True. S is not open or closed.
9) S is a closed set.
- False. S is not closed because it does not contain its boundary point 1.
10) T is neither open nor closed set.
- False. T is a closed set.
11) 0 is an accumulation point of set S.
- False. No open ball centered at 0 contains any points of S.
12) T does not have any accumulation point.
- False. i and 1 are accumulation points of T.