Why is "topological sorting" called "topological"? Is it just because it determines an order without altering any vertices or edges -- like a doughnut and coffee cup are topologically equivalent? Why is it not called "dependency sort" or something else? Why "topological"? I admit I'm mystified.
The earliest reference I could find for topological sort is from [Lasser61]:
A network of directed line segments free of circular elements is assumed. The lines are identified by their terminal nodes and the nodes are assumed to be numbered by a non-topological system. Given a list of these lines in numeric order, a simple technique can be used to create at high speed a list in topological order.
I don't have access to this article right now but I would wager that the "topology" in "topological sort" does not come from the mathematical notion of topology (e.g.: open sets, compactness, etc...) but rather from the "network topology" sense.
[Lasser61] Lasser, Daniel J. "Topological ordering of a list of randomly-numbered elements of a network." Communications of the ACM 4, no. 4 (1961): 167-168.
The topology of a set of items is how they are connected. Topological sorting is sorting items based only on their topology.