The minimal cardinality where the Reznichenko property fails

A topological space X has the Fréchet-Urysohn property if for each subset A of X and each element x in Ā, there exists a countable sequence of elements of A which converges to x. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect each neighborhood of x. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a set X of real numbers such that C_p(X) does not have the weak Fréchet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space ^ℕℕ.) We prove the Kočinac-Scheepers conjecture by showing that if C _p(X) has the Reznichenko property, then a continuous image of X cannot be a subbase for a non-feeble filter on ℕ.