IF S is the set of all x such that 0<= x <= 1, what points, if any, are points of accumulation of both S and the compliment of S, C(S).
I don't get this question, the answer looks like 0 and 1 but can somebody explain please !!!
0 and 1 would be accumulation points for the set 0<x<1, but this isn't the set you gave. If it is an accumulation point, this point isn't in the set, and the points you gave are inside of the set, so those can't be accumulation points. There are no accumulation points for S. Now think of the compliment.
The part in red is not correct.
If $\displaystyle S=(0,1)\text{ or }S=[0,1]$ then in either case both $\displaystyle 0 \&~1$ are accumulation points of $\displaystyle S~\&~S^c$.
If $\displaystyle \delta > 0$ then the open set $\displaystyle (-\delta,\delta)$ contains points of both $\displaystyle S~\&~S^c$ different from $\displaystyle 0$,
With respect to 1, the same is true of $\displaystyle (1-\delta,1+\delta)$