Mathematical Economics

We have provided with the geometric realizations of open and closed balls in  endowed with any of the three "commonly examined" metrics,  showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric. Their geometric relations were later on exemplified via the notion of metric equivalence and its balls' implications. We thus examined the antimonotonic relationship between balls of the same centered arising by pairs of dominant and dominating metrics.

We begun our study of metric properties with the finitary notion of boundedness. 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. 

We have begun to show several general poperties of the notion and provide with examples: e.g.  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. 

Notes on the above can be found here, here, here, here. 

The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found  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.