View source for Lebesgue number

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==Definition==
Let {{M|(X,d)}} be a [[metric space]], and {{M|\mathcal{U} }} be a [[open cover]] of {{M|X}}. We define the ''Lebesgue number''{{rITTMJML}} as follows:
* if there is a {{M|\delta\in\mathbb{R} }} such that {{M|\delta>0}} such that {{M|1=\forall A\in\mathcal{P}(X)\ \exists U\in\mathcal{U}[\text{Diam}(A)<\delta\implies A\subseteq U]}}, then {{M|\delta}} is the ''Lebesgue number'' for {{M|\mathcal{U} }}.
In words: a number, {{M|\delta>0}}, is called a ''Lebesgue number'' for the cover {{M|\mathcal{U} }} if for every subset of {{M|X}} whose [[diameter (metric space)|diameter]] is {{M|<\delta}} is contained in one of the {{M|U\in\mathcal{U} }}.
: '''Recall:'''
:* [[Diameter (metric space)|Diameter]] - the ''diameter'' of a [[bounded set]], {{M|A}} in a [[metric space]], {{M|(X,d)}} is defined as:
:** {{M|1=\text{Diam}(A):=\text{Sup}(\{d(x,y)\ \vert\ x,y\in A\})}} (where {{M|\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''
==References==
<references/>
{{Definition|Metric Space|Topology}}