Lebesgue number

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Definition

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

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

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

Recall:

• Diameter - the diameter of a bounded set, [ilmath]A[/ilmath] in a metric space, [ilmath](X,d)[/ilmath] is defined as:
  • [ilmath]\text{Diam}(A):=\text{Sup}(\{d(x,y)\ \vert\ x,y\in A\})[/ilmath] (where [ilmath]\text{Sup} [/ilmath] 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