Definition 1: "A mathematical structure is nothing but a (more or less) complicated organization of smaller, more fundamental mathematical substructures. Numbers are one kind of structure, and they can be used to build bigger structures like vectors and matrices."
Definition 2: 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.
Sets: "A Set is a well-defined collection of objects."
When the author says "structure on set" or "attach structures to sets", what does it mean? How can I "attach" structures to sets?
"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."
My thoughts on what "attach" is:
If I build a set, and the elements of the Set satisfies required axioms of Algebraic Structure for Ring Set then I can say that I have successfully attached a structure (Algebraic Structure (Ring Set here)) to the built set?