Volume 1
Summer 2006
Number 1

Pushing The Limit: How Far Can We Go?

(generalized limits and limit extrema in topology)

Andrew James Critch

Abstract: The traditional epsilon-delta definition of a limit is for functions in  metric spaces.  This paper defines limits for relation in topological spaces, and agrees with the usual definition  in the original cases.  The advantage is that less information is needed and more interesting cases are covered, as the functional dependence on a "variable" is unnecessary for evaluating the limits of many expressions (for example, the limit of a Riemann sum as the norm of its partition approaches zero: the relation between the norm and the sum is not functional, yet the limit is naturally defined).  The general definition is in some ways simpler, providing enlightening perspectives on traditional uses of limits, and a context for the generalization of limit extrema (limsup and liminf) in unordered spaces via a concept of "limit closure."