Why are homeomorphisms important? I attended a guest lecture (I'm in high school) hosted by an algebraic topologist. Of course, the talk was non-rigorous, and gave a brief introduction to the subject. I learned that the goal of algebraic topology is to classify surfaces in a way that it is easy to tell whether or not surfaces are homeomorphic to each other. I was just wondering now, why are homeomorphisms important? Why is it so important to find out whether two surfaces are homeomorphic to each other or not? 

The notion of homeomorphism is of fundamental importance in topology because it is the correct way to think of equality of topological spaces. That is, if two spaces are homeomorphic, then they are indistinguishable in the sense that they have exactly the same topological properties.
