TITLE: geometry and topology
QUESTION [16 upvotes]: I was wondering what are the differences and relations: 
between geometry and topology; 
between differential geometry and differential topology;
between algebraic geometry and algebraic topology?
For example:
Are they studying different objects? Such as different mathematical structures/spaces? 
if they study the same object, but study different aspects/properties of the same object?
...
Reading their wikipedia pages really confuses me.
Thanks and regards!

REPLY [2 votes]: I have the same question when I heard of the word Topology during I was studying how to render a mesh in computer. Here is my understanding now for the difference between Topology and Geometry. Without losing of generality, take a triangular mesh as an example because spaces/complexes can find a triangulation.
Topology is a structure or a framework between the elements that can be found on a complex(e.g. a 2D-surface. It is no doubt that the complex's skeleton is a set of elements too(e.g. vertex, edge, face). I always keep in mind that Topology is a studying of neighborhood for Geometry. This was what I knew during very beginning. Then construction of spaces, manifold...etc are more advanced topic.
Geometry is study of the realization of the skeleton. Realizations are maps from the abstract manifold space concept to your real life $R^3$. The simplest would be the triangular mesh that has been widely used for many industries. The realizations are plane equations for each face->triangle. All skeletons exist in the same space simultaneously.