When is examining only one direction of a bijective map enough to categorize it as an isomorphism?
Let X,Y be sets with being bijective. If we consider X and Y various structures and ask what conditions do we have to impose on for it to be an isomorphism, we can break these cases up into two groups:
- If X,Y are vector spaces, then needs to be linear.
- If X,Y are groups, then needs to respect the group operation.
- If X,Y are metric spaces, then needs to be an isometry.
- If X,Y are topological spaces, then needs to be continuous and open.- If X,Y are smooth manifolds, then needs to be a diffeomorphism.
In case one, we only need to verify that satisfies certain properties (that is linear, respects group operation, isometry, etc.). Whereas in group 2, we need to verify that both AND satisfy certain properties (continuous, smooth, etc). My question is, what is it that distinguishes the members of group 1 verses group 2? Why is it enough to check a certain condition in one direction only enough in certain cases?