nLab induced metric

 Idea

A subset of a metric space canonically inherits a metric by restriction. This is called the induced metric.

The corresponding metric topology is the subspace topology.

Examples

• The Euclidean metric on an Archimedean ordered field is an induced metric. Every Archimedean ordered field is an ordered subfield of the Dedekind real numbers and thus inherits the Euclidean metric of the Dedekind real numbers.

• The Euclidean metric on a real interval is an induced metric, since real intervals are subsets of the Dedekind real numbers.