Lets A,B topological spacestopologies T1 y T2 respectively
A function
F : A → B

Is said to be a homeomorphismif F is continuous and bijective and also F-1 exists and is continuous and bijective

Homeomorphism concept refers to F preserves the topology for spaces A and B and they are topological equivalent, F being continuous implies that

∀ U∈ T1, ∃ V∈T2 : U= F-1(V)

or equivalently

∀ε >:0, ∃ δ >0 : |x - y| <δ => |F(x) - F(y)| < ε