12:27 PM
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If we have any directed set $(D,\le)$ then we can add a point $\infty\notin D$ and then consider the topology on $C(D)=D\cup\{\infty\}$ such that all points of $D$ are isolated and the local base at $\infty$ consists of all "upper sets" $\langle d,\infty\rangle=\{\infty\}\cup\{d'\in D; d'\ge d\}$...

Do you know tha treatment of convergence in Preuss Topologische Räume I? It's even more general and IMO quite elegant. No mention of your $D$ though. For sequences we have one space to deal with and for nets a huge class of such spaces (characterised by being almost discrete, i.e. having exactly one non-isolated point). — Henno Brandsma 4 hours ago
I have some trouble which books is meant here.
The Wikipedia article Gerhard Preuß (Mathematiker) mentions Allgemeine Topologie and Foundations of Topology - An Approach to Convenient Topology - I have access to both these books.