A convergence not metrizable

TL;DR

This paper investigates the conditions under which different notions of convergence of sequences of functions are metrizable, highlighting cases where pointwise convergence is not equivalent to metric convergence.

Contribution

It provides insights into the topological properties of convergence notions, especially demonstrating non-metrizability of pointwise convergence under certain conditions.

Findings

Pointwise convergence can be non-metrizable under specific hypotheses.

Topologies for convergence on compact or bounded sets are often metrizable.

The paper clarifies the relationship between various convergence notions and their topological properties.

Abstract

Certain notions of convergence of sequences functions such as pointwise convergence and (uniform) convergence on compact or bounded sets come from suitable topological function spaces; see [1]. Under certain conditions these topologies involved are metrizable, which in an advantage since there is an extensive theory on convergence in metric spaces. However, the case of pointwise convergence is delicate, since it is shown that under certain hypotheses this form of convergence of sequences of functions is not equivalent to convergence in metric.