Return to 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 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 $\mathrm{CC}(\beta(r)) \subseteq \mathrm{CC}(r) \subseteq \mathrm{CC}(\gamma(r))$.

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

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

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

This question arised 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 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 \mathcal{P} 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 $\mathcal{P} U$ 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 $\mathrm{CC}(\beta(r)) \subseteq \mathrm{CC}(r) \subseteq \mathrm{CC}(\gamma(r))$.

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