Some thoughts while I learn point-free topology...

General topology generalizes the concept of closeness from metric spaces. With general topology, we do not need to actually know the distance to talk about points being close to a set (closure) or converging to other points (getting arbitrarily close). We do not need the distance to talk about a certain function being or not continuous. This is GENERAL TOPOLOGY.

Some very smart people have realized that we do not actually need to talk about points to do general topology. This is POINT-FREE TOPOLOGY!