Mathematical Economics

Given the tutorial preparation of the notion of continuity of functions between metric spaces and a brief re-examination of this during the lecture, we have provided with a major 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 criteria.

You can find here notes for the above. We note that the topology generated by the uniform metric might be unecessary strong for the examination of the asymptotic behavior of sequences of maximizers, and a weaker topology for this, is the one of hypo-convergence.

We briefly examined the non-topological notion of completeness of a metric space by first intrducing the notion of a Cauchy sequence inside a metric space. You can find notes for this, here.

Exercise: Try to show that the result on the continuity of the sup operator holds when this is defined over the whole set of bounded functions.