In Fantasy math, Peter Saveliev remarks:
Removing structures from consideration is hard; adding is much easier. Compare how you start with point-set topology -- by removing the geometry from the Euclidean space -- and how you start with inner product spaces -- by adding this structure to vector spaces.
Questions: Has research been carried out on the difference in difficulty, if any, between removing and adding structures when it comes to learning a mathematical concept or topic? If so, what were the conclusions? Is it inherently easier to think about additional structure on a familiar object, and inherently harder to let go of already familiar structures? Are any aspects of this special to mathematics or is it merely an example of a more general phenomenon of difficulty in unlearning what you have already learned?
(Mathematical examples and counterexamples are welcome.)
Note: while I am using the term "structure" somewhat loosely, it may be helpful to see the Wikipedia article on mathematical structure(s). I quote the opening sentences below:
In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.