This is not true, but I'll leave the counterexample to you (it's not difficult).every subset of a compact set is compact

For the idea is to assume there exists a sequence with no accumulation points in , then there exist intervals with center at that contain (from ) only (why?). Clearly is closed (why?) so it is compact (why?), but can the cover have a finite subcover? (I picked a sequence because it's easier to visualize, but T need not be countable)

For just use Heine Borel.