Course : Mathematical Economics

After some brief discussion of the course's scope and aims, as well as of some prerequisite notions, we begun with the definition of a distance function (metric) w.r.t. 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, so

We have been occupied with Brouwer's FPT and its subsequent application for the verification of the existence of Nash equilibria in a class of finite non-co-operative games, in a restricted setting that avoids the use of its generalization to correspendences, i.e. Kakutani's FPT.  You can find notes on the above here.

We have been occupied with the development of preparatory notions for the Brouwer's FPT. You can find notes for them here. (Notice that the sketch of proof of Borsuk's Lemma is out of the scope of the lectures. Anyhow, the notes contain a correction of the sketch as presented in the class. Specifically, the isomorphism between the homology groups of the same order does not hold for retracts in general, but refers to the stronger notion of deformation retracts. Check the notes for the corrected a

We have been occupied with issues involving the Banach FPT, generalizations and applications involving the properties of the Bellman equation and Picard's Theorem. Yoy can find notes on the above here.

We have continued the examination of the non-topological notion of completeness of a metric space. You can find here notes for it. We have also studied the stronger than topological continuty notion of Lipschitz continuity, obtaining finally the notion of a contraction. You can find here notes for it. 

We continued the examination of issues involving convergence and continuity. After a little bit of trouble, we have provided with an application that establishes the continuity of the sup functional when properly restricted to possibly non-empty metric subspaces of the set of bounded real functions equiped with the uniform metric. Using this and under further assumptions we were occupied with the issue of convergence of approximate maximizers under uniform convergence of their respective criteri

We continued the study of the concept of sequential convergence. We have proven that it can be equivalently described by systems of open neighborhoods, hence it is a notion present in more general topological spaces. We have proven that a sequence inside a metric space can have at most one limit, due to the property of separation. The consideration of the behavior of sequences in indiscrete topological spaces implied that there exist topologies not generated by metrics. We discussed the issue of

We continued with the examination of the notion of total boundness. We proved that it is a refinement of the notion of boundness, in the sense that total boundeness implies boundness, while we refered to their equivalence in Euclidean spaces. We were occupied with the issues of  the equivalence of metrics w.r.t. total boundness, and the invariance of the notion w.r.t. finite products.

We begun the examination of topological notions on a metric space that arise by the presence of a metric. We def

We have been occupied with issues of hereditarity on subsets of boundness, equivalence of metrics w.r.t. boundness, and the "invariance" of boundness w.r.t. finite products and the metrics we have defined on such products. We moved on to a refinement of the notion, obtaining the notion of total boundness, and we have initially provided with the non-equivalence of the two notions.

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,  they can separate points in a metric space, and the local information that they convey about their center, can be conveyed by a "countable description". The example of the real line endowed with the usual metric and the real line endowed with the discrete one showed that the “form” of the