Mathematical Economics

We continued the study of the concept of sequential convergence. We have proven that it can be equivalently described by systems of open neighborhoods, hence it is a notion present in more general topological spaces. 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. We discussed the issue of characterization of topological properties via sequential convergence. 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.

We begun the examination of continuity of functions "between" metric spaces. We provided of a sequential characterization of continuity at a point, as a property that does not destroy convergence to the point. We also provided equivalent characterizations via open balls and systems of open neighborhoods. The latter characterization implies that this notion is more generic, in that it is definable for functions between general topological spaces.