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.
This solution goes over mathematical concepts within the realm of geometry and topology, such as compact and closed topologies.