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arXiv · 1703.07327

When is U(X) a ring?

Javier Cabello Sánchez
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Abstract

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space $(X,d)$ is a ring if and only if every subset $A\subset X$ has one of the following properties: $A$ is Bourbaki-bounded, i.e., every uniformly continuous function on $X$ is bounded on $A$. $A$ contains an infinite uniformly isolated subset, i.e., there exist $δ>0$ and an infinite subset $F\subset A$ such that $d(a,x)\geq δ$ for every $a\in F, x\in X$.

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Javier Cabello Sánchez. 2017-03-21. When is U(X) a ring?. https://doi.org/10.2298/fil1707981c

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