Three modifications of connectedness

Question

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 connectors $\gamma (r)$ and $\beta (r)$ 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 $\mathrm{CC}(\beta(r)) \subseteq \mathrm{CC}(r) \subseteq \mathrm{CC}(\gamma(r))$.

Answer 1

First let me note that I agree with many of the comments and my answering the question should not be taken as disagreement. In particular, I find something off-putting about asking such a question on a topic about which you are writing a paper. It would be one thing if this were a mere curiosity (questions of the form "does such and such approach lead anywhere" come up a lot, and can be ok if phrased right). This seems more of the form "I don't feel like checking the details; could you prove my lemmas or give me counterexamples". Nonetheless I had some time and felt like playing around with some random idea which has nothing to do with the conference I'm currently at (on game theory). So I'm answering against my better judgement.

Define the disjointness relation $\delta$ by $A\delta B \Leftrightarrow A\cap B = \emptyset$. Here are some easy things to check:

1) If $r$ is a connector then so is $r\cap\delta$ and $CC(r) = CC(r\cap\delta)$. (Here I'm assuming "connector" means a relation which satisfies the monotonicity property you specified; that wasn't entirely clear)

2) $r\subseteq\gamma(r)$ and $\gamma(r)\cap\delta\subseteq r\cap\delta$, so $\gamma(r)\cap\delta = r\cap\delta$.

3) $CC(\gamma(r)) = CC(\gamma(r)\cap\delta)=CC(r\cap\delta)=CC(r)$

As for the question of $\beta(r)$, it should be noted that as you defined it (or, as I'm assuming you were meaning to define it) $\beta(r)$ is not a connector. Suppose it were. Then $A\in CC(r)$ implies $A\beta(r)\emptyset$. By the definition of a connector, $B\beta(r)\emptyset$ whenever $A\subseteq B$, so $B\in CC(r)$. That is to say, any superset of a connected set is connected. This is usually not true; say, if $r$ defines the usual connectedness of a discrete space. So it's not clear that it makes sense to ask about $CC(\beta(r))$, because as a set it may not have the properties you'd expect for a notion of connectedness.

Answer 2

From the definition of $\beta(r)$ it's obvios that $CC(\beta(r))=CC(r)$.

$CC(\beta(r))=\{A\subseteq U\ |\ \forall X,Y\subset A,\ X\not=\emptyset\not=Y \land X\cup Y=A \land X\cap Y=\emptyset$ $\implies X\cup Y=A\in CC(r)\}=\{A\subseteq U\ | \ A\in CC(r)\}$.

Now assume $A\in CC(\gamma(r))$ and $X,Y\subset A$ non empty such that $X\cup Y=A$ and $X\cap Y=\emptyset$, then $X\gamma(r) Y$. So $\exists N\subset X \land M\subset Y$ such that $NrM$. Now if $r$ is extendable, we have $XrY$. So we can conclude that $A\in CC(r)$.

Is this what you were asking?