A previous question Covering the plane with compact sets received a (to me) unexpected, simple, and elegant solution, which, moreover, possessed the merit of continuity. This time the constraints are more demanding.
The problem is to find two partitions of the plane into closed sets, related to each other as follows. The first partition consists of pairwise disjoint closed sets $V_{\alpha}$ all of which except one, $V_0$, are compact. The second partition likewise consists of pairwise disjoint closed sets $W_{\beta}$ all of which except one, $W_0$, are compact. The partitions are related thus:
(1) $V_0$ and $W_0$ are disjoint;
(2) each $V_{\alpha}$ (except $V_0$) meets each $W_{\beta}$ (except $W_0$) in exactly a singleton;
(3) each $V_{\alpha}$ meets $W_0$ in at most a singleton, and each $W_{\beta}$ meets $V_0$ in at most a singleton.
Can two such partitions be found?