A set $B\subset \Bbb{R}$ is a Bernstein set if it intersect every perfect subsets of $\Bbb{R}$.
Why Bernstein sets are useful?
Any set with positive outer measure contains a non measurable subset.
Any set of second category contains a set that lacks the property of Baire.
We use Bernstein set to prove the existence of non measurable set with the property of Baire and a measurable set without baire property ( given here).
Bernstein sets are "big " in sense of cardinality (uncountable), in sense of measure (positive outer measure) and in sense of Baire category (second category) but the difference set $d(B)=B-B$ contains no interval.
We can prove that every closed set is the union of a countable set and a perfect set using the point of condensation. Hence every closed uncountable set intersect a perfect set.
Can we prove that every closed uncountable set intersect a perfect set using Bernstein set ?
Can we prove that every non empty perfect set has cardinality $\mathfrak{c}$ using Bernstein set ?
What are other interesting applications of Bernstein set?
