PS 7

  1. [0,infinity) - if the space is not compact, then produce an open cover that has no finite subcover. Otherwise, prove that it is compact.
  2. X with the cofinite topology - if the space is not compact, then produce an open cover that has no finite subcover. Otherwise, prove that it is compact.
  3. Give an example of a bounded metric space that is not compact. Show that the space is bounded and produce an open cover that has no finite subcover.
  4. Prove that the intersection of two compact sets in a Hausdorff space X is compact in X.
  5. Prove that Hausdorff is required in the above statement, ie that the intersection of compact subspaces of a space X is not necessarily compact as follows:
          Look at Y=[0,1] U [2,3] with the equivalence relation ~ on Y s.t.    t ~ t for all t,    t ~ t+2 for all t in [0,1),    t ~ t-2 for all t in [2,3)
          Show that Y/~ is not Hausdorff
          Show that [0,1] U [2,3) is compact in Y/~
          Show that [0,1) U[2,3] is compact in Y/~
          Show that the intersection of these two compact sets [0,1) U [2,3)in Y/~ is not compact in Y/~
Extra Credit In class we proved that [0,1] is compact, but as part of our proof, it looks like we proved that [0,1) is also compact - contradicting that [0,1-1/n) is a cover with no finite subcover. Explain in detail what goes wrong for the proof when you try and modify it for [0,1) as the original space.