Mathematical Economics

Given the general definition of a topological space we defined the concept of a metrizable topology (i.e. a topology generated by a metric). We derived an easy example concerning the indiscrete topology that showed the existence of topologies that are not metrizable. We were occupied with the issue of when two metrics define the same topology using the usual concepts of comparisons between metrics.

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 property of separation. The consideration of the behavior of sequences in indiscrete topological spaces implied that there exist topologies not generated by metrics. Hence we have proven that closeness of a subset is equivalent to that every convergent sequence with elements in the subsets converges inside the subset. While closeness implies this sequential property in every topological space, the converse is true in metric spaces due to first countability. We were occupied with the issue of when two different metrics imply the same asymptotic behavior for the same sequence using again the usual notions of comparison.

You can find notes for the above here.