## Exercises on Compactness

- [0,infinity) - if the space is not compact, then
produce an open cover that has no finite subcover. Otherwise, prove
that it is compact.
- 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.
- Cantor set - if the space is not compact, then produce and open cover that
has no finite subcover. Otherwise, prove that it is compact.
- 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.
- Prove that the intersection of two compact sets in a Hausdorff
space X is compact in X.
__Grad__
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/~