I am currently self studying algebraic topology from the book "topology and groupoids".
I don't understand classifying spaces up to homotopy type. By "I don't understand" I don't mean that I don't understand the proof of facts about homotopy types that are in my book. These facts follow easily from the definitions and I did not face any trouble in proving them. By not understanding, I mean that I don't realize why this is being done.
I know that homeomorphic spaces are of the same homotopy type. So it seems to me that homotopy type is some sort of a topological invariant. However, it is not clear to if determining whether 2 spaces are of the same homotopy type is really easier than determining if the 2 spaces are homeomorphic. Determining whether 2 spaces are both connected seems an easier problem to me than determining if they are homeomorphic.
I think I will be much more satisfied, if someone can show me an example of showing that 2 spaces are not homeomorphic by showing that they are not of the same homotopy type.
Thanks for your time.