Apologies for the length of the question.
Consider interval $I=[0,1]$. For any $n \in \mathbb{N}$ we can always find two finite sets $S_{1n} \subset I$ and $S_{2n} \subset I$ such that:
a) $S_{1n}\cap S_{2n} = \emptyset$
b) There is at most one point from $S_{1n} $ between any two consecutive points of $S_{2n}$. Similarly, there is at most one point from $S_{2n} $ between any two consecutive points of $S_{1n}$.
c) $H(S_{1n}, I) \rightarrow 0$ and $H(S_{2n}, I) \rightarrow 0$ as $n \rightarrow \infty$, where $H$ stands for the Hausdorff distance.
E.g., we can always choose: $$S_{1n}=\left\{ \frac{k}{2^n}: \, k - \text{ odd }, \; 0 \leq k \leq 2^n\right\},$$ $$S_{2n}=\left\{ \frac{k}{2^n}: \, k - \text{ even }, \; 0 \leq k \leq 2^n\right\}.$$
Here is why property b) is interesting to me. For any given $n$, it holds that from any point $a \in (0,1)$ there is always a direction (right or left) to go where the first encountered point is going to be a point from $S_{1n} $. Similarly, there is always a direction (right or left) to go where the first encountered point is going to be a point from $S_{2n}$.
A construction like this is not possible with three disjoint sets on $I$.
$\star$ $\star$ $\star$
However, I wonder if something analogous is possible with three sets when we consider the square $I_2=[0,1]^2$.
In a nutshell, I wonder if for $n \in \mathbb{N}$ we can always find three finite lattices $S_{1n} \subset I_2$, $S_{2n} \subset I_2$ and $S_{3n} \subset I_2$ such that:
A*) Their projections $S^x_{1n}$, $S^x_{2n}$, $S^x_{3n}$ on the first coordinate are pairwise disjoint. Their projections $S^y_{1n}$, $S^y_{2n}$, $S^y_{3n}$ on the second coordinate are pairwise disjoint.
B*) With any point $b=(b_1,b_2)\in \bigcup_{j=1}^3 S_{jn}$ associate a “cross” $$\hat{b} = (b_1\times [0,1])\cup ([0,1]\times b_2).$$ Then for any $n$ construct the net from these “crosses”: $$W_n= \bigcup_{b \in S_{1n} \cup S_{2n} \cup S_{3n}} \hat{b} $$
Here is a description of the property analogous to b). For any given $n$ and for any $j=1,2,3$, it holds that from any point $a \in Int(I_2)$ there is always a direction (either horizontal or vertical – so we have four possible directions now) to go where the first encountered point on $ W_n $ is going to be from a “cross” associated with a point from $S_{jn}$.
C*) $H(S_{jn}, I_2) \rightarrow 0$as $n \rightarrow \infty$, $j=1,2,3$.