I have some questions, but not sure if they are meaningful:
- Suppose $X$ and $Y$ are two arbitrary measurable spaces. Does there exist a measurable mapping from $X$ to $Y$?
- Suppose $X$ and $Y$ are two arbitrary measure spaces. Does there exist a measure-preserving mapping from $X$ to $Y$?
- Suppose $X$ and $Y$ are two arbitrary topological spaces. Does there exist a continuous mapping from $X$ to $Y$?
They are similar in this way:
if $X$ and $Y$ are two arbitrary sets with some type of structure, does there exist a structure-preserving mapping from $X$ to $Y$?
Hope to see if there can be some insights.
Thanks and regards!