I was thinking about nets the other day, when I was reminded of something that I wondered when I first encountered them. Is the generalization to partially-ordered sets strictly necessary? The generalization to partially-ordered sets is useful, but are there spaces with points that are not reachable by totally-ordered nets alone?
In a first countable space, a point lies in the closure of a set if and only if there is a sequence of points in the set that converges to the point. This property fails in general. For example, an uncountable set with the cofinite topology has no non-trivial convergent sequences, but the closure of any infinite set is the whole space. You can recover this property if you pass to nets, which allow fairly general partially-ordered sets to be the index set (the only requirement you must impose is that they be directed sets). So if you require the index sets of your nets to be totally ordered, is there a space which contains a point that is not the limit of such a net?
Poking around Wikipedia, I found that the page for order topology, which suggests that the Tychonoff plank is an example of a space where totally-ordered nets are not sufficient. The page discusses nets indexed by ordinals, which possibly is a loophole, but it seems like a very narrow one. I’d be curious if anyone knows for sure.