This question arose in my research of generalized connectedness (see this draft article for the overall idea, but beware that the draft is yet too preliminary and unreadable, however I hope you can understand the overall idea from the draft):

Let $U$ is a set, $r$ is a binary relation on $\mathcal{P} U$. I call $r$ a connector.

Informal note: The relation $r$ of two sets $A$ and $B$ represents that $A$ and $B$ are in some sense "near" or "touch". For example $r$ may be a proximity.

I will call a connector $r$ extendable when

$ \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) . $

Below I will require that $r$ is extendable.

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 \mathcal{P} A \setminus \lbrace \emptyset \rbrace : (X \cup Y = A \wedge X \cap Y = \emptyset \Rightarrow X r Y) \rbrace . $

As I mentioned above, $r$ may be a proximity and in this case $\mathrm{CC}(r)$ is proximal connectedness, that is a set a $A$ is connected iff every partition of the set a $A$ consists of two near sets.

As an other important example $ArB$ may mean that the topological closure (given some topological space) of the set $A$ in the subspace generated by the set $A\cup B$ intersects $B$ or the closure of $B$ intersect $A$. This is equivalent to the classic definition of connectedness of a set on topological space, because it happens if and only if $A$ and $B$ are not both open-closed on $A\cup B$.

There are other examples of connectedness following this scheme: graph connectedness, digraph strong connectedness, uniform connectedness, etc. (see my draft article)

I will define binary relations $\gamma (r)$ and $\beta (r)$ on the set $\mathcal{P} U$ 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 the above mentioned draft article I proved that $\mathrm{CC}(\beta(r)) \subseteq \mathrm{CC}(r) \subseteq \mathrm{CC}(\gamma(r))$.