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The minimal cardinality where the Reznichenko property fails

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Abstract

A topological spaceX has the Fréchet-Urysohn property if for each subsetA ofX and each elementx inĀ, there exists a countable sequence of elements ofA which converges tox. 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 ofx. In [5], Kočinac and Scheepers conjecture:

The minimal cardinality of a setX of real numbers such thatC 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 spaceNℕ.) We prove the Kočinac-Scheepers conjecture by showing that ifC p(X) has the Reznichenko property, then a continuous image ofX cannot be a subbase for a non-feeble filter on ℕ.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Boaz Tsaban

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  1. Boaz Tsaban
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The author is partially supported by the Golda Meir Fund and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).

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Tsaban, B. The minimal cardinality where the Reznichenko property fails. Isr. J. Math. 140, 367–374 (2004). https://doi.org/10.1007/BF02786640

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  • DOI: https://doi.org/10.1007/BF02786640

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