irreducible compact set vs prime compact set

Question

Let $X$ be a topological space. A compact set $K$ is called irreducible if for any two compact subsets $K_1$, $K_2$ of $K$ with $K$ is equal to the union of $K_1$ and $K_2$, then $K$ is equal to $K_1$ or $K_2$. A compact set $K$ is called prime if for any two compact sets $K_1$, $K_2$ of $X$ with $K$ is included in the union of $K_1$ and $K_2$, then $K$ is included in $K_1$ or $K_2$.

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

Answer 1

OK, it seems that here is a construction.

As the OP claims, there exists an infinite irreducible compact $T_0$-space $Y$. Consider any its minimal open cover $S_1,S_2,\dots,S_n$ with $n\geq2$ (it necessarily exists!). We may assume that $n=2$ by merging $S_2,\dots,S_n$.

Now construct the space $X$ by adding to $Y$ two new points $x_1,x_2$ and adding to a base of the topology two sets $T_i=S_i\cup\{x_i\}$; this does not change the induced topology on $Y$, so $Y$ is still an irreducible compact.

On the other hand, each of the $T_i$ is compact, as any open set containing $x_i$ covers $T_i$. These two compacts cover $Y$, but none of them covers it separately. So $Y$ is not prime.