A common theme is that there is an "Object/Structure" to which "Maps" are applied; after which invariants can be studied.
In topology, it's topological spaces/structures whose maps are continuous functions (or continuous deformations). Generally my understanding is that, using the topology of the underlying space, we can understand what "continuity" means there, and as such use those maps that are continuous for further study; the simplest case being homeomorphisms which preserve the topology of the structure.
I also understand that metric spaces are special cases of topological ones. Here we introduce a metric function and the notion of distance becomes available. My issue is, at what point do things become more geometric and less topological? One can argue that geometric spaces are built on top of topological ones (but is it always the case?). Sure, geometric spaces can have richer features (inner products/angles, metric/distance, and even measures).
But at what point can we say, this is geometric, uses topology, but isn't purely topology? My thinking is "topology introduces continuity" and "geometry introduces measurement". What scares me with these statements is that there are non-metric geometries (such as projective geometry); is it still considered measuring anything at all? I also understand that there are axiomatic approaches to geometries that might not really follow the "geometric structure + geometric transformations" approach.
In essence, are geometric spaces just richer topological ones? Or are they fundamental in their own right?
Please correct me if in some areas my reasoning is inaccurate or just plain false. I want to learn more delicate ways to look at things. Also, please throw any ideas you have to improve my train of thought, I would like more experienced minds to shed light.
Thank you!
