Talk:Union of Overlapping Convex Sets in Toset is Convex/Infinite Union

There's a very close analogy to connectedness here. The proof I'm writing of this theorem will also work, with only tiny changes, to prove a similar theorem about connectedness (using the fact that a space is connected iff any two points in it are connected). I've been trying unsuccessfully to factor out the common elements as a lemma I guess for now I'll just try to get both proofs written. --Dfeuer (talk) 04:01, 14 February 2013 (UTC)

Improve
I feel in my bones like there's a better way to do this, but it keeps eluding me. The relation on elements of $\mathcal A$ defined by
 * $P \sim Q$ iff there is a finite sequence of elements of $\mathcal A$ linking them together

is an equivalence relation.

In particular, it's the transitive closure of the "has non-empty intersection with" relation.

The relation on $\bigcup \mathcal A$ defined by $a \sim' c$ iff $a < b < c \implies b \in \bigcup \mathcal A$ is also an equivalence relation. It turns out to be the transitive closure of "lies in the same element of $\mathcal A$ as".

I want to somehow put these ideas together in some nice neat fashion that will also give a framework to prove the version of this theorem for connected sets, but it just keeps eluding me. Help? --Dfeuer (talk) 20:45, 14 February 2013 (UTC)