This question arised in my research of generalized connectedness:

Let $U$ is a set, $r$ is a binary relation on $U$. I will require

$ \forall X_0, Y_0, X_1, Y_1 \in \mathcal{P} U : (X_1 \cap Y_1 = \emptyset \wedge X_0 r Y_0 \wedge X_1 \supseteq X_0 \wedge Y_1 \supseteq Y_0 \Rightarrow X_1 r Y_1) . $

I will define the set $\mathrm{CC} (r)$ of connected subsets of $U$ by the formula

$ \mathrm{CC} (r) = \lbrace A \in \mathcal{P} U | \forall X, Y \in A \setminus \lbrace \emptyset \rbrace : (X \cup Y = A \wedge X \cap Y = \emptyset \Rightarrow X r Y) \rbrace . $

I will define binary relations $\gamma (r)$ and $\beta (r)$ on the set $U$ by the formulas (for every $A, B \in \mathcal{P} U$)

$ A \gamma (r) B \Leftrightarrow \exists X \in \mathcal{P} A, Y \in \mathcal{P} B : X r Y. $

$ A \beta (r) B \Leftrightarrow A \cup B \in \mathrm{CC} (r) . $

Conjecture: $\mathrm{CC} (\gamma (r)) = \mathrm{CC} (\beta (r)) = \mathrm{CC} (r)$.

If it is wrong I want to see the counter-examples with which it fails and additional condition under which it is indeed true.

In this draft article I proved that $\mathrm{CC}(\beta(r)) \subseteq \mathrm{CC}(r) \subseteq \mathrm{CC}(\gamma(r))$.