Assignment:
Let S be the set [0,1] and define a subset F of S to be closed if either it is finite or is equal to S.
- Prove that this definition of closed set yields a topology for S.
- Show that S with this topology is compact, but S is not a Hausdorff space.
- Show that each subset of S is compact and that therefore there are compact subsets of S that are not closed.
Provide complete and step by step solution for the question and show calculations and use formulas.