Mathematical Economics

After a presentation of the course's scope and aims, and using the overview of the familiar case of the real numbers, we begun with the definition of a distance function (metric) over a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families of metrics, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space.  

Procedural details can be found at the course's syllabus, notes on the previous here, and a counterexample of a metric here. The whiteboards from the analogous lecture of the Ac. Year 2020-21 (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Exercise: Using as carrier set the real line, provide a counterexample of a function that fails to be a metric because it fails to satisfy the triangle inequality, even though it satisfies the remaining properties of the definition.