How do I show, I am assuming by example, that X can be homeomorphic to a subspace of Y and Y is homeomorphic to a subspace of X but X and Y are not homeomorphic?

I have tried things involving the indiscrete and the discrete topologies as these seemed simple to work with, however I have been unable to come up with anything.

Edit: I have added an extra question later on in the thread that I am stuck on.