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.
By the way, all the rest of the argument the Hallsofivy is true. Closed in compact space (i.e closed intersected compact) is always compact, in every topological space, as it is discussed in other threads. Hence this exercise needs to show that there is a non closed compact in this space. Further, they are needed two of these compacts for intersecting them, because closed and compact would give compact . I suppose the non discrete topology in {0,1} is that whose unique open sets are all the space and the empty set. Thus we have to take benefit from the fact that neither or are open nor closed in endowed with the trivial topology.
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
I believe both are compacts. is homeomorphic to . Any sequence or net in this space is convergent if and only if is. Notice that is always convergent to both 0 and 1. Hence, roughly speaking,
"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.