Mathematics itself is largely "all about definitions". I once had a student ask about a problem in the text (Linear Algebra). I read the problem out to the class, wrote a single word from the problem on the chalk board, and asked "what is the definition of that word". I got a lot of blank stares so I sat down at my desk, placed my hands on the desk, and waited. Eventually, the students started leafing through the book and finally one even looked up the word in the index! Once they had the precise definition of the word, they could do the problem.
Definitions in mathematics are "working definitions"- you use the precise words of the definition in proofs and problems.
As Plato said, that is rather crudely put. A "basis" for the product topology is the collection of all sets of the form where U is in X and V is in Y. But then we must be able to take finite intersections and unions which can give open sets that are NOT of that form. For example, the open disk, is open in even though it is not of the form " " for any open U and V in R (which would be an open rectangle).