Lebesgue number

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Definition

Let $(X,d)$ be a metric space, and $\mathcal{U}$ be a open cover of $X$. We define the Lebesgue number as follows:

• if there is a $\delta\in\mathbb{R}$ such that $\delta>0$ such that $\forall A\in\mathcal{P}(X)\ \exists U\in\mathcal{U}[\text{Diam}(A)<\delta\implies A\subseteq U]$, then $\delta$ is the Lebesgue number for $\mathcal{U}$.

In words: a number, $\delta>0$, is called a Lebesgue number for the cover $\mathcal{U}$ if for every subset of $X$ whose diameter is $<\delta$ is contained in one of the $U\in\mathcal{U}$.

Recall:

• Diameter - the diameter of a bounded set, $A$ in a metric space, $(X,d)$ is defined as:
  • $\text{Diam}(A):=\text{Sup}(\{d(x,y)\ \vert\ x,y\in A\})$ (where $\text{Sup}$ denotes the supremum of a a set of real numbers)

See also

• Lebesgue number lemma - every open cover of a compact metric space has a Lebesgue number