Mathematical Economics

We continued our study of ball properties and have shown that the local information that they convey about their center, can be conveyed by a "countable description", as well as that there exists an antimonotone relation between the open (closed) balls of fixed center and radius w.r.t. metrics that obey functional inequalities. 

We begun our study of metric space properties with the finitary notion of boundness. The existence of the collection of the open balls in a metric space allows for the definition of the concept of a bounded subset (via an obvious extension of the analogous definition in the real line w.r.t. the usual metric). It holds iff it can be covered by an open (equivalently closed) ball, while the center of the covering ball need not be an element of the subset at hand. We have shown that any finite set is (universally) bounded, while the open and closed balls are by construction bounded. The notion is hereditary. If a set is bounded then any subset is also bounded. The dual is evident. If a set is not bounded then any superset is also unbounded. Obviously the notion depends crucially on the metric. For example the discrete space is always bounded. This is in contrast with any Euclidean space (i.e. the  equipped with the usual metric) which is not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one, this also implies that the metric space consisting of the set of bounded real functions on X with the uniform metric is not generally bounded, etc. 

You can find notes for the above here and here.