I'm trying to prove one final result for my thesis on cardinal functions on hyperspaces, which states that for a LOTS (linearly ordered topological space) $X$, we have that $nw(X) = w(X)$.
I have already proven this for compact spaces, locally compact spaces and metrizable spaces, but after a long time have failed to conjure a proof for this one. Interestingly, this result is often cited to Engelking's General Topology, 3.12.4(d), where it's left as an exercise for the reader.
Any pointers? Or for that matter, proofs?
See lemma 3 on page 34 of this paper by Hajnal which shows that for $X$ a LOTS, $w(X) = d(X) + g(X)$, where $g(X)$ is the number of "gaps" of $X$, i.e. open intervals $(u,v)$ with $u < v$, that are empty, so that two points are "direct neighbours". The proof of $w(X) \le g(X) + d(X)$ is not that hard: let $G$ be the set of all gap-endpoints (so for each $u < v$ with no $x$ with $u < x < v$, $u,v \in G$, and let $D$ be dense of minimal size, then the set of isolated points of $X$ plus all open intervals with endpoints in $G \cup D$ form a base for $X$, by some case distinguishing).
I think the proof of $w(X) \ge g(X) + d(X)$ can also be used to see that $nw(X) \ge g(X)$ (and $nw(X) \ge d(X)$ is clear), and then $w(X) \ge nw(X) \ge d(X) + g(X) = w(X)$ and equality ensues. This is clear: let $N$ be any network for $X$, then for each gap $(u,v)$, the set $(u,\rightarrow)$ is open and contains $v$ so $\exists N_v$ such that $v \in N_v \subseteq (u, \rightarrow)$ and it follows that $v = \min(N_v)$. So there are as least as many distinct members of $N$ as there are gaps.
For a connected LOTS, $w(X) = d(X)$ is easy to see, and $nw(X)$ sits between those two cardinals, so that's a boring special case.