The Wikipedia's article for geometry is somehow overwhelming. To make things clear, allow me to ask some questions:
I wonder if "geometry" can be defined as the study of a metric space (possibly with or without other structures)?
Any thing more general than metric space (such as uniform spaces and topological spaces) is not in the scope of "geometry"?
Does "geometry" assume the set under study to have some algebraic structure?
Also there is algebraic geometries.
Is the underlying set a topological vector space, normed space, inner product space, or even Euclidean space?
Since projective and affine spaces are pure algebraic concepts without metrics, why are there "projective geometry" and "affine geometry"?
Thanks and regards!