The minimal cardinality where the Reznichenko property fails

Abstract

A topological space X$ has the Frechet-Urysohn property if for each subset A of X and each element x in the closure of A, 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 all neighborhoods of x. In their paper, Kocinac and Scheepers conjecture that the minimal cardinality of a set X of real numbers such that C_p(X) does not have the weak Frechet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space). We prove the Kocinac-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 the natural numbers.