The Minimal Cardinality Where the Reznichenko Property Fails
نویسنده
چکیده
A topological space X has the Fréchet-Urysohn property if for each subset A of X and each element x in 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 each neighborhood of x. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a set X of real numbers such that Cp(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 N). We prove the Kočinac-Scheepers conjecture by showing that if Cp(X) has the Reznichenko property, then a continuous image of X cannot be a subbase for a non-feeble filter on N.
منابع مشابه
On the Pytkeev Property in Spaces of Continuous Functions
Answering a question of Sakai, we show that the minimal cardinality of a set of reals X such that Cp(X) does not have the Pytkeev property is equal to the pseudo-intersection number p. Our approach leads to a natural characterization of the Pytkeev property of Cp(X) by means of a covering property of X, and to a similar result for the Reznichenko property of Cp(X).
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