In my understanding, a bijective map establishes a one-to-one correspondence between elements, and has an inverse. This represents that the map garantees topologically equivalent transformation. (*homeomorphism == bijective, continuous inverse)
If a topological structure is transformed(mapped) with a bijective function to the other topological space, then the transformed structure is topologically equivalent with the original one like donut == mug cup.
I just want to (intuitively) understand how the bijective map garantees the topological equivalence in topological structure mapping. How do I know that the transformed structure is topologically equivalent with the original one?
How does the one-to-one mapping garantee the topological equivalence, conceptually?