What is Metric (mathematics)?

In mathematics, a metric or distance function is a function that gives a distance between each pair of point elements of a set. A set with a metric is a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is a metrizable space.

One important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor allows distances along curves to be determined through integration, and thus determines a metric.