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Suppose $E\subset R$,if $E$ can be covered by a series of interval $\{I_{\lambda}\}_{\lambda\in \Lambda}$,then E can be covered by a countable subset series which are in $\{I_{\lambda}\}_{\lambda\in \Lambda}$ ?

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    Not correct. Consider the intervals $\{[a,a]\}_{a\in R}$. They cover $R$ but don't have a countable sub-cover.2012-09-24
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    you are right.Actually I should add up that the intervals are non-empty interiors.2012-09-24

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