up:: Metric Space
It is defined as the set of all Open Sets in Metric Spaces, i.e. the set of all sets whose points have some Open Balls (Metric Spaces) also interior to . Thus, The metric topology is generated by open balls.
Note that Topological spaces are generalizations of metric space topologies; that is, these properties of metric topologies motivated the more general definition of Topological Spaces.
Note that, when seen as topological spaces (with their metric topologies), All metric spaces are Hausdorff spaces, since all points are separated by a distance of, say, .