Compact refinement of a covering

Question

Suppose $X$ is compact. $A$,$B$ open sets which cover $X$. Can $X$ be covered by compact sets $C$,$D$ such that $C \subseteq A$ and $D \subseteq B$?

Answer 1

Let $X = \alpha(\mathbb{Q})$ denote the Alexandroff compactification of $\mathbb{Q}$. It is a compact non-Hausdorff $T_1$-space which is moreover a KC-space (which means that all compact subsets are closed). See the answers to https://math.stackexchange.com/q/2793610.

Now assume that $X$ satisfies the "compact refinement" property of your question. Then $X$ would be normal since it is a KC-space. But among compact $T_1$-spaces, Hausdorff is equivalent to normal. This is a contradiction.

Answer 2

We need normality of $X$ (so $X$ Hausdorff is sufficient too), because the fact that every two-set open cover has a two-set closed refinement is equivalent to normality. (easy proof by taking complements.)