Mathematical Economics

We have completed our basic vocabulary regarding balls in metric spaces by showing that those can be used in order to seperate points. This property (which does not generally hold for pseudo-metrics) is essential, since it among others provides with the uniqueness of limits in metric spaces.

We begun our study of metric properties with the finitary notion of boundness. The balls can be readily used in order to define it as a natural extension of the notion of boundness on the real line (w.r.t. the usual metric). Specifically a subset of a metric space is bounded  iff it can be covered by an open (equivalently closed) ball. The center of the covering ball need not be an element of the subset at hand although we have shown that when such a ball exists the center can always be chosen to lie inside this subset.

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 (i.e. universally bounded), etc. We have examined the notion of uniform boundness and provided with (counter-) examples.

You can find notes for the above here and here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.