METRIC SPACE

in mathematics, a set together with a rule, called a distance function, for relating a number to every pair of objects from the set such that the correspondence has properties similar to those of distance in Euclidean space. That is, the distance function (dist) is always positive and satisfies the triangle inequality (that is, the length of any side of a triangle is less than the sum of the other two sides) for any three points. Any metric space can be made into a topological space by choosing the open subsets of the topology to be the analogues of circles in the Euclidean plane; that is, for each point p and each positive real number r the corresponding open set consists of all points x such that dist(p, x) is less than r. One important characteristic of metric spaces is that they have the Hausdorff property, meaning that sequences can have only one limit point. Metrization is a process for making a suitable Hausdorff topological space into a metric space in such a way that the open sets resulting from the use of the distance function, as in the preceding paragraph, are the same as the open sets of the original topology. The determination of exactly which types of topological spaces are metrizable is one of the important problems in general topology.