# Alternative characterization of connectivity spaces.

First, a barrage of definitions.

Define that a relation on a set $X$ is a subset of $X^2$. If $f$ is a symmetric relation on $X$, define that $f$ is connected if and only if there do not exist disjoint and exhaustive $A, B \subseteq X$ and non-empty relations $\alpha,\beta$ on $A$ and $B$ respectively such that $f = \alpha \cup \beta$.

Given a collection of sets $\mathcal{J}$, define the that overlap relation is the set $$\mathcal{J}^{\mathrm{ov}} = \{(A,B)|A,B \in \mathcal{J}, A \cap B \neq \emptyset\}.$$

Finally, define that a connectivity space is a pair $(X,\mathcal{K})$ with $\mathcal{K} \subseteq \mathcal{P}(X)$ such that

1. $\emptyset \in \mathcal{K}$

2. For all non-empty $\mathcal{J} \subseteq \mathcal{K}$ such that $\bigcap \mathcal{J} \neq \emptyset,$ it holds that $\bigcup \mathcal{J} \in \mathcal{K}$.

(Note: It is typical to call the elements of $\mathcal{K}$ "connected sets." Also, I presume that the set of all connected relations on a set $X$ forms a connectivity space; however, this is tangential to the issue.)

Question: Suppose we replaced condition 2 in the definition of a connectivity space with the following.

2.* For all non-empty $\mathcal{J} \subseteq \mathcal{K}$ such that $\mathcal{J}^{\mathrm{ov}}$ is connected, it holds that $\bigcup \mathcal{J} \in \mathcal{K}$.

Is the resulting definition equivalent to the original?

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Every relation satisfies your definition of "connected" -- just take $A = \emptyset$, $B = X$. And the definition feels like it's describing something disconnected. Furthermore, you don't use the definition in the rest of your question. Are you sure you've transcribed everything correctly? –  Amit Kumar Gupta Feb 26 '13 at 6:19
Thanks, thats probably why i was finding the question so hard! Can you look over my "fixed" definition and let me know if it works? –  goblin Feb 26 '13 at 6:23
Sorry, I edited my comment since I thought you weren't using the definition of connected relation, but now I see you use it in reference to $\mathcal{J}^{ov}$. Unfortunately I can no longer edit my comment. –  Amit Kumar Gupta Feb 26 '13 at 6:26
Its no big deal. Anyway, i feel bad that i screwed up the question... –  goblin Feb 26 '13 at 6:29

First we show $(2^\ast) \rightarrow (2)$:

Assume $(2^\ast)$, and let $\mathcal{J} \subset \mathcal{K}$ be arbitrary, but such that $\bigcap \mathcal{J} \neq \emptyset$. We can conclude $\bigcup \mathcal{J} \in \mathcal{K}$ if we can show that $\mathcal{J}^{ov}$ is connected. We're assuming that $\bigcup \mathcal{J} \neq \emptyset$, so $\mathcal{J}^{ov} = \mathcal{J}^2$, i.e. the "full" relation on $\mathcal{J}$. It's not hard to see that this relation is connected; let's extract this as a:

Lemma: if $S \neq \emptyset$ is a set, then $S^2$ is connected.
Proof: Suppose not, and let $A, B, \alpha, \beta$ witness this. $A, B$ are non-empty, so let's pick elements from them, say $a, b$. Clearly $(a,b) \in S^2$, but it's impossible for $(a,b) \in \alpha \cup \beta$ since $\alpha \subset A^2$, $\beta \subset B^2$, and $A, B$ are disjoint from one another.

Now $(2) \rightarrow (2^\ast)$:

Let $\mathcal{J}$ be such that $\mathcal{J}^{ov}$ is connected. We'll apply Zorn's Lemma to see that $\bigcup \mathcal{J}\in\mathcal{K}$. Let $$\mathbb{P} = \left\{ \mathcal{I}\subset\mathcal{J}\ |\ \bigcup \mathcal{I}\in\mathcal{K}\right\}$$

And partially order $\mathbb{P}$ by inclusion. $\mathcal{P}$ is not empty: for any $J \in \mathcal{J}$, $\mathcal{I} = \{J\}$ witnesses the non-emptiness. Every chain in $\mathbb{P}$ has an upper bound in $\mathbb{P}$: this follows from your observation about nested collections of connected sets, which follows immediately from $(2)$. Thus by Zorn's Lemma $\mathbb{P}$ has a maximal element. Let's pick one and call it $\mathcal{I}$. I claim that $\mathcal{I} = \mathcal{J}$ (and so $\bigcup\mathcal{J}\in\mathcal{K}$, as desired).

Suppose not. Case 1: there is some $J\in\mathcal{J}\setminus\mathcal{I}$ such that $J\cap\bigcup\mathcal{I}\neq\emptyset$. If that were the case, then by $(2)$, $\{J\}\cup\bigcup\mathcal{I}\in\mathcal{K}$, but then $\{J\}\cup\mathcal{I}\in\mathbb{P}$, contradicting the maximality of $\mathcal{I}$. Case 2: Every $J\in\mathcal{J}\setminus\mathcal{I}$ is disjoint from $\bigcup\mathcal{I}$. This also yields a contradiction, for $A=\mathcal{I}$, $B=\mathcal{J}\setminus\mathcal{I}$ witness that $J^{ov}$ is disconnected.

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Are you sure its false? Consider three pairwise overlapping sets whose mutual inersection is empty. According to both definitions, their union is an element of $\mathcal{K}$. Its immediate from 2*, but slightly non-trivial from 2. –  goblin Feb 26 '13 at 7:00
Good point. The counterexample I had in mind is wrong; but it made me think of a better one. If you have collection $\mathcal{J}\subset\mathcal{K}$ of $n$ sets, each intersecting the next so they form a "chain", then $(2^\ast)$ forces their union to belong to $\mathcal{K}$ too. Does $(2)$ force the same fact? As you suggest, yes, but non-trivially. First apply $(2)$ to get that the union of the first two of the sets is in $\mathcal{K}$. Apply $(2)$ again to get that the union of the first three is in. Continuing you get $\bigcup\mathcal{J}\in\mathcal{K}$. Now what if $\mathcal{J}$ is infinite? –  Amit Kumar Gupta Feb 26 '13 at 7:24
Good point. Perhaps adding closure with respect to unions of nested collections would fix the problem... –  goblin Feb 26 '13 at 7:41
As in, the union of a nested collection of connected sets is a connected set. –  goblin Feb 26 '13 at 7:43
Yup, you got it. And so $\alpha = A^{ov}$, $\beta = B^{ov}$. –  Amit Kumar Gupta Feb 26 '13 at 10:50