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Reference.org
Bernstein set
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<p> In mathematics, a Bernstein set is a subset of the <a href="/facts/Real_line/6eq42xX5">real line</a> that meets every uncountable <a href="/facts/Closed_subset/upYnRTUj">closed subset</a> of the real line but that contains none of them. </p><p>A Bernstein set <a href="/facts/Partition_of_a_set/VeP9g8CA">partitions the real line</a> into two pieces in a peculiar way: every <a href="/facts/Lebesgue_measure/8us9cGoW">measurable</a> set of positive measure meets both the Bernstein set and its <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a>, as does every set with the <a href="/facts/Property_of_Baire/G2Sk5Azz">property of Baire</a> that is not a <a href="/facts/Meagre_set/AxTk8frw">meagre set</a>. </p>