Basic to all mathematics is the notion-here used quite informally-of a set with structure. For every type of structure there is a notion of equivalence (or isomorphism)-a one-to-one onto function that, in an appropriate sense, preserves the structure. A particular type of structure defines a branch of mathematics: the study of those concepts preserved by equivalence. For example, a group is a set furnished with the structure group operation. The notion of equivalence is the usual notion of isomorphism of groups. (Semi-Riemannian geometry with applications to Relativity, by Barrett O’neill)
Consider following table:
I know if $f:X\to Y$ be a one-to-one onto function, then two sets $X$ and $Y$ have the same cardinality and it is sufficient for me to visualize the equivalence in this case.
If $f:X\to Y$ be a homeomorphism, then one can deform the topological space $X$ to the topological space $Y$ without cutting and gluing, it is sufficient for me to visualize the equivalence in this case. If a coffee cup and a donut are given to me, I can realize from their shapes that they are homeomorphic.
If $f:X\to Y$ be an isometry, then one can coincide the semi-riemannian manifold $X$ to the semi-riemannian manifold $Y$, it is sufficient for me to visualize the equivalence in this case.
Question 1: Is the above statement true?
Question 2: I have no idea to visualize the equivalence in the Manifold theory case, diffeomorphism. Can someone help me? Can I judge about the equivalence of the two manifolds from their shapes in this case?