Independent Bernstein sets and algebraic constructions

We present the method of constructing algebras and linear spaces of 2c2c generators using independent Bernstein sets. As an application we obtain large algebras of special functions: (i) strongly everywhere surjective functions which are not perfectly everywhere surjective; (ii) nowhere continuous Darboux functions; (iii) nowhere continuous compact to compact functions; (iv) functions which are continuous precisely on a fixed closed proper subset of RR. Most conclusions obtained in this paper are improvements of some already known results.