Mathematical Economics

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. boundness, and here a result on boundness obtained by some asymptotic comparison of balls between two metrics.

We moved on to a refinement of the notion, obtaining the notion of total boundness. If we can perceive boundness as equivalent to the existence of some ε>0 for which there exists a finite cover of open (or equivalently closed) balls for the set at hand, total boundness strengthens this by requiring that for any ε>0 there exists an analogous finite cover. By definition, for any ε>0, the analogous cover would be essentially determined by the finite set of ball centers, that a. can depend on ε, and b. need not lie inside the set at hand. We begun working on the notion by establishing that the version of the definition involving covers consisting of open balls is equivalent to the version of the definition involving covers of closed balls.

We continued establishing first that total boundness implies boundness, and then moved on to examine issues of hereditarity, of the universal total boundness for finite sets, of the definition of a totally bounded space, the subsequent construction of counter-examples, and then showed that the notion is generally stronger (as expected) than the usual boundness, by showing that a discrete space is totally bounded iff it is finite, and by providing a(n) (applicationwise more interesting) example of a subset of a function space that is uniformly bounded but not totaly bounded w.r.t. the uniform metric.

You can find notes for the above here.