Let X$X$ be a topological space. A compact set K$K$ is called irreducible if for any two compact subsets K1$K_1$,K2 $K_2$ of K$K$ with K$K$ is equal to the union of K1$K_1$ and K2$K_2$, then K$K$ is equal to K1$K_1$ or K2$K_2$. A compact set K$K$ is called prime if for any two compact sets K1$K_1$,K2 $K_2$ of X$X$ with K$K$ is included in the union of K1$K_1$ and K2$K_2$, then K$K$ is included in K1$K_1$ or K2$K_2$.
Are these two properties equivalent? I guess that they are not the same. But I could not come up with anan example.
