Limits. Continuity

Abstract

As mentioned in the introduction, the limit concept is one of those at the origin of topology. The student already knows several aspects of this concept: limit of a sequence of points in a metric space, limit of a function at a point, etc. To avoid a proliferation of statements later on, we present in §2 a framework (limit along a ‘filter base’) that encompasses all of the useful aspects of limits. It doesn’t hurt to understand this general definition, but it is much more important to be familiar with a host of special cases.