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Say I have some set $E\subset \mathbb{R}$, and an open cover $\cup A_i \supset E$. Is there a general algorithm that makes this cover disjoint?

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    In general, an open cover of $E$ does not have a disjoint subcover. What is meant by making "this" cover disjoint?2011-04-15
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    I imagine you intended $E\subset R$, not $E \in R$.2011-04-15
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    In addition to what user6312 wrote, you may want to look at the [Vitali covering lemma](http://en.wikipedia.org/wiki/Vitali_covering_lemma).2011-04-15

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