Article Dans Une Revue Asian Research Journal of Mathematics Année : 2025

Incomplete Metrics on \(\mathbb{R}\) with Usual Topology

Résumé

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.

Dates et versions

hal-05054892 , version 1 (03-05-2025)

Identifiants

Citer

Tonny K B. Incomplete Metrics on \(\mathbb{R}\) with Usual Topology. Asian Research Journal of Mathematics, 2025, 21 (5), pp.147-153. ⟨10.9734/arjom/2025/v21i5930⟩. ⟨hal-05054892⟩
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