Mathematical Economics

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, obey some monotonicity property,  they can separate points in a metric space, and the local information that they convey about their center, can be conveyed by a "countable description". The example of the real line endowed with the usual metric and the real line endowed with the discrete one showed that the “form” of the open (and/or the closed) balls crucially depends on the metric.

A finitary property that the existence of balls in a metric space make examinable, is the 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. The open and closed balls are essentially considered bounded and the notion is also considered as hereditary. The center of the covering ball need not be an element of the subset at hand. 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.). Obviously 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.