Many subtle definitions, such as interior point, accumulation point, open set, closed set, are all based on the concepts of distance and neighborhoods.  The beauty is mathematics is you could always go further with good understanding of the fundamental ideas by going into general. The metric space is a good example of the generalization of ideas, from distance to metric . Further to the metric space. Most concepts which have been defined under the framework of distance and neighborhood thus can be extended to this more general setting. This is one way mathematics extended to more general setting and developed into new math branch.