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