Mathematical Economics

We continued the examination of the notion of total boundness. We provided several remarks regarding hereditarity, etc, that implied that the notion coincides with that of boundness on Euclidean spaces. We briefly discussed the notion of metric entropy and described the Metric Entropy Integral Theorem of Dudley as an astonishing application of the notion of total boundness in the theory of Gaussian Processes. As previously you can find notes for total boundness here and for the aforementioned remarks and discussions see also here (notice that the latter contains the correction of the system of inequalities for the covering numbers w.r.t. the particular equivalence between different metrics compared to what was derived in class).

Exercise: Prove that the metric appearing in Dudley's Theorem is actually a pseudo-metric (and notice that the notions of boundness and total boundness can be readily extended to pseudo-metrics).

We begun the examination of topological notions on a metric space that arise by the presence of a metric. We initially defined the notion of a topology on a non-empty set and subsequently the notion of open and closed subsets as duals. You can find notes for the above here.