Except unions of open sets are required to be open, so the case for topology here is not what you're describing (assuming standard definitions).

Generally, the topology on a graph will look like what you get if you draw your graph in the euclidean plane (or space, whatever) without intersections of edges.

The notion of closeness of vertices in this case isn't really well described by "point set topology", but you'd rather use the notion of distance between vertices (length of a shortest path). But even then, the distance stuff generally assumes non-directed edges, because you generally want the distance from A to B to be the same as from B to A.

In short, I don't think "point set topology" has much to say here, at least as usually done.

You're right. You can use the more general mathematical topology to describe a network topology. It's just mathematical topology speaks in sets, compactness, closeness that people not visualize well. The simpler and more specific network topology or network graph can be visualized very well.