Incomplete Metrics on \(\mathbb{R}\) with Usual Topology
Tonny K B *
Department of Mathematics, College of Engineering Trivandrum, Kerala 695016, India.
*Author to whom correspondence should be addressed.
Abstract
The set of all real numbers \(\mathbb{R}\) with the usual metric is complete, and it generates the usual topology on \(\mathbb{R}\). Motivated from the fact that incomplete metric on \(\mathbb{R}\) that produces usual topology on \(\mathbb{R}\) provides some insight about the relation between topological and metric structure of \(\mathbb{R}\), present paper discusses the existence of an incomplete metric on \(\mathbb{R}\) that generates the usual topology on it. The paper demonstrates that completeness is a metric property, not a topological one. We proved some general results that lead to a method to identify infinitely many incomplete metrics on \(\mathbb{R}\). Moreover, the existence of such incomplete metrics on \(\mathbb{R}\) highlights the presence of metrics on \(\mathbb{R}\) which are not norm induced.
Keywords: Metric space, complete metric, incomplete metric, topology