Course : Mathematical Economics

During the second TA session we were proeccopied with the following:

First, we revisited and corrected exercise 8 of the problem set 1 that was erroneously solved during the first TA session (find the solutions here).

Secondly, we proved two lemmas. The first concerns total boundness and finite product spaces, already seen in class. The proof is analogous to the proof of boundness and finite products. The second concerns the equivalence of characterising continuity using open balls or neigbo

We were occupied with the comparison of topologies generated by different by different metrics that obey functional inequalities.

We begun the study of the concept of sequential convergence given the notion of a neighborhood system of a point in a topological space and we have proven that in a metric space the notion can be equivalently described via the subsystem of open balls centered at the limit. We have proven that a sequence inside a metric space can have at most one limit, due to the prop

We concluded our examination of the notion of total boundness by considering issues involving metric comparisons and finite products. We provided several remarks regarding hereditarity, etc, that implied that the notion coincides with that of boundness on Euclidean spaces.

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 open and closed subsets as duals, then the subsequent notion of a topology arising fr

On our first TA session we went through two sets of exercises on some elementary notions of metric spaces, also covered during the lectures. In particular we tried to firmly embed and understand the properties of metric functions and open (closed) balls in metric spaces.

You can find here and here solutions to all the exercises that we saw during the TA session. As mentioned in class, the rest of the exercises are either similar to those we solved or will not preoccupy as in general.

As was

We have been occupied with further examples, and the issues of 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. You can find notes for the above here and here.

We have also been occupied with an example of a "sequence of metric spaces", where the property is "somehow lost in the limit", that can be found here. You can also find here a generalization of the issue of metrics comparison w.r.t. boundne

We continued our study of ball properties and have shown that the local information that they convey about their center, can be conveyed by a "countable description", as well as that there exists an antimonotone relation between the open (closed) balls of fixed center and radius w.r.t. metrics that obey functional inequalities. 

We begun our study of metric space properties with the finitary notion of boundness. The existence of the collection of the open balls in a metric space allows for the d

You can find here and here exercises corresponding to the notions that were examined in the first three lectures.

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 monot

We have been occupied with further examples of metrics and subsequent metric spaces, and have shown that it is possible that different metrics on the same carrier set can obey relations. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as with a motivation of further examination of such relations.

We examined the important example of the space of bounded real functions on a non-empty domain endow

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, po