A Lebesgue number of an open cover of a subset

Question

This topic has been absolutely beaten to death, but I was trying to precisely define a Lebesgue number for an open cover of a subset of a metric space and I ended up confusing myself.

Let $M$ be a metric space and let $S \subset M$. Let $\mathcal{U}$ be an open cover of $S$ (open w.r.t. the metric topology). I want to define a Lebesgue number of $\mathcal{U}$ (and whether or not it exists at all) in the usual way. Namely:

A Lebesgue number of $\mathcal{U}$ is a number $\delta>0$ such that every subset of $S$ having diameter less than $\delta$ is contained in some member of $\mathcal{U}$.

My question is: How should I define what counts as an "open cover" in this case? In particular, should I let members of the open cover include elements outside of $S$, or should I constrain it to only include subsets of $S$? I have (confusingly) seen open covers of subsets defined both ways. I'm not sure if there just isn't any agreement, or what I've read has been wrong.

Most definitions for Lebesgue number I have seen are for open covers of the entire space $M$. In that case, obviously there is only one way to go about things. However if I try to carry that definition over to a subset by considering the subset as the entire space, then... that's what generates the confusion.

I'd appreciate anything to clear up the water here. Thanks!

Answer 1

What you may be missing is the concept of the "subspace topology", also known as the "relative topology". This is the topology on the subset $S$, each of whose open sets has the form $U \cap S$ where $U$ is any open subset of $M$. Based on this, it does not matter whether you use open subsets $U$ of $M$, or their intersecions $U \cap S$, because for any subset $A \subset S$ we have $A \subset U \iff A \subset S \cap U$.