Mathematical Economics

We completed our general definitions with the notion of the metric subspace. This as well as  the notion of product metric spaces (with a finite number of factors) can be viewed as ways to construct further metric spaces from given one(s), with the resulting metrics carrying relevant information on the given one(s).

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, and obey some monotonicity property. The examples of the real line endowed with the usual metric, the real line endowed with the "exponential metric", the real line endowed with the discrete one, or of the  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. We also suspected that the inclusion relations between  the balls (with the same center and radius) in  w.r.t. the three "commonly examined" metrics may be connected to the functional relations that we have previously derived between the metrics.

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 confirming the aforementioned remark concerning . We also provided a set theoretic relation between the balls (open or closed) of fixed center and radius of a metric space and a metric subspace of the former. 

You can find notes for the above here and here.