Metric spaces
Metric Spaces I. Basic Notions

In any topological space, there is generally no tool for measuring the proximity between two points (p. 78 of ), unlike in metric spaces, where this was the original aim.

The notion of metric space was formally introduced by M. Fréchet (1906). A metric space is a set for which the distances between its elements are rigorously defined using a distance function whose properties are the abstraction and generalization of those possessed by historical Euclidean distance.