Let X be the cartesian product of with the usual topology and a two point set with the indiscrete topology. Find two compact subspaces of X such that their intersection is not compact.
Can I get some help please?
That's not possible. A compact set is closed in any topology. The intersection of two closed sets is closed in any topology. A closed subset of a compact set is compact in any topology. Therefore, the intersection of two compact sets is compact is always compact no matter what topology you have.
I guess these two subsets
are compact, and the intersection
clearly is not (is isomorphic to (0,1]).
I suggest proving the compactness of these subsets
"anything" is compact iff is compact in . The argument of an open recovering
also work because is always the whole space. I think is compact if and only if the projection is compact in .
Anyway I prefer people to check this, non Hausdorff spaces are not familiar to me, but I like the problem.