What is the difference between an autohomeomorphism and a bijection?
Suppose we work on topological spaces.
Bijection = function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
Homeomorphism = A homeomorphism f of topological spaces is a continuous, bijective map such that its inverse is also continuous.
Autohomeomorphism (also known as automorphism or self-homeomorphism) = homeomorphism from a topological space to itself.
So is my understanding correct that homeomorphism is like a bijection, but additionaly it preserves the structure of topological space? Does my diagram show correct thinking about this? What are other differences between these notions to keep in mind?
Thank you very much!
Best Answer
Yes your understanding is correct,
If there exist an homeomorphism between two topological spaces you can say that the two spaces are the same when you see it as topological spaces.
Homeomorphism is a special case of a more general type of map : the isomorphism, it is a bijective map that preserve the structure, if you need to specify the structure isomorphism can have different name.
for example : an isomorphism between two topological spaces is a homeomorphism
an isomorphism between two smooth manifolds is a diffeomorphism
an isomorphism between two metric spaces is an isometry