# Preservation of uniform continuity under pointwise product

## TL;DR

This paper investigates conditions under which the space of uniformly continuous functions on a metric space remains stable when multiplied pointwise, providing various characterizations that extend to broader contexts.

## Contribution

It offers new characterizations of the stability of uniformly continuous functions under pointwise multiplication, extending known results to more general settings.

## Key findings

- Characterizations of stability under pointwise product for $U(X)$
- Extensions of results to broader classes of spaces
- Conditions ensuring $U(X)$ remains closed under multiplication

## Abstract

Let $X$ be a uniform space and $U(X)$ the linear space of real-valued uniformly continuous functions on $X$. Our main objective is to give a number of properties characterizing the fact that $U(X)$ is stable under pointwise product in case $X$ is a metric space. Some of these characterizations hold in much more general circumstances.