General topology is a pretty far cry from geometry though. What you need for the reals isn't even directly related to the topology of R, since you just need tha

General topology is a pretty far cry from geometry though. What you need for the reals isn't even directly related to the topology of R, since you just need that every set with an upper bound has a least upper bound, which is a property that many types of orders can have and which doesn't necessarily have anything to do with geometry.