Find two different topologies on where one is strictly finer, yet the two are homeomorphic
Let and be bases for topologies and .
is strictly finer than , since for each x in Z and for each basis element containing x, we have a basis element in that contains x. The converse is not necessarily true.
Since there is a bijection between Z and Q, we have a homeomorphism between and as subspaces of R with a discrete topology.