Mathematical Economics

We completed our general definitions with the notions of the metric subspaces and of the product metric spaces (with a finite number of factors). We can view the above 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, obey some monotonicity property,  and they can separate points in a metric space, while this does not generally hold for pseudo metrics. 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.

You can find notes for the above here and here.