Let X be a topological space. A compact set K is called irreducible if for any two compact subsets K1,K2 of K with K is equal to the union of K1 and K2, then K is equal to K1 or K2. A compact set K is called prime if for any two compact sets K1,K2 of X with K is included in the union of K1 and K2, then K is included in K1 or K2.

Are these two properties equivalent? I guess that they are not the same. But I could not come up with an example.