Mathematical Economics

Total boundness is a refinement of boundness. The latter is equivalent to the existence of some ε>0 for which there exists a finite cover of open (or equivalently closed) balls for the set at hand. Total boundness strengthens this by requiring that for any ε>0 there exists an analogous finite cover. By definition, for any ε>0, the analogous cover would be essentially determined by the finite set of ball centers, that a. can depend on ε, and b. need not lie inside the set at hand (but we can force them to-how?).  

Total boundness is important as it is connected to the determination of a notion of complexity of the space involved, something that is particularly handy when we study spaces comprised of functions. Even though it is not a topological notion, later on we will see that it is related to compactness.

You can find notes for the above here and here.