Clarification on the definition of the Lebesgue number of a metric space Definition: Let $X$ be a metric space and $\mathcal{O}$ an open cover of $X$.  A Lebesgue number for $\mathcal{O}$ is a positive number $\varepsilon$ with the property that every subset of $X$ of diameter less than $\varepsilon$ is contained in some member of $\mathcal{O}$.
What specifically does 'some' mean?  Does it mean a single member of $\mathcal{O}$, or could it mean that the subset could lie entirely within the intersection of multiple members of $\mathcal{O}$?  The given wording doesn't make this clear to me.
The reason I ask is the use of the Lebesgue number in the proof of the covering path property for the 1-sphere, where we choose a Lebesgue number $\varepsilon$ for the open cover $\{\alpha^{-1}(U_1), \alpha^{-1}(U_2)\}$, where $U_1$ is an open arc on the 1-sphere extending counterclockwise from $(-1,0)$ to $(0,1)$, $U_2$ is an open arc extending counterclockwise from $(1,0)$ to $(0,-1)$, and $\alpha:I \to S^1$ is a path with initial point $(1,0)$. Taking a finite sequence $0=t_0 < t_1 < \cdots < t_n=1$ with successive terms differing by less than $\varepsilon$ we have that $\alpha([t_i,t_{i+1}])$, $0 \le i \le n-1$ must be contained in either $U_1$ or in $U_2$.
Am I correct in thinking that the above implies the subset cannot lie in more than one open set?

In the general case, you just can say that there exists at least one member of the covering $\mathcal{O}$ containing that subset.
There is no problem if the covering $\mathcal{O}$ contains two open sets with one of them contained on the other.
