Some topics related to metrics and norms, including ultrametrics and ultranorms

TL;DR

This paper explores geometric properties of metrics and norms, focusing on ultrametrics and ultranorms, particularly in the context of vector spaces over fields with possibly non-archimedean absolute values.

Contribution

It investigates the topological and geometric characteristics of ultrametrics and ultranorms, highlighting their differences from classical metrics and norms in vector spaces.

Findings

Ultrametrics exhibit strong non-archimedean properties.

Topological dimension zero is analyzed in relation to these metrics.

Connections between connectedness and ultranorms are established.

Abstract

Here we look at some geometric properties related to connectedness and topological dimension 0, especially in connection with norms on vector spaces over fields with absolute value functions, which may be non-archimedian.