Definition (vid). A function is a homeomorhism if it is continuous, bijective, and its inverse is also continuous. Some comments
To prove two things are homeomorphic, it's enough to find a mapping which is a homemorphism. However, to prove two things are not homeomorphic you have to prove that no mapping will be a homemorphism! Because there may be too many maps to check, one most often has to find a topological invariant, that is different in the two spaces, to prove this.
Some examples of topological properties/invariants (Topological property):
A continuous Open function is a homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar and μορφή (morphē) = shape, form.[1]. See wiki