This is an announcement for the paper "Pointwise convergence of partial functions: The Gerlits-Nagy Problem" by Tal Orenshtein and Boaz Tsaban.
Abstract: For a set X of real numbers, let B(X) denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product R^X. Assume that for each countable subset A of B(X), each f in the closure of A is in the closure of $A$ under pointwise limits of sequences of partial functions. We show that in this case, B(X) is countably Frechet-Urysohn, that is, each point in the closure of a countable set is a limit of a sequence of elements of that set. This solves a problem of Arnold Miller. The continuous version of this problem is equivalent to a notorious open problem of Gerlits and Nagy. Answering a question of Salvador Hernandez, we show that the same result holds for the space of all Baire class 1 functions on X. We conjecture that the answer to the continuous version of this problem is negative, but we identify a nontrivial class of sets X of real numbers, for which we can provide a positive solution to this problem. The proofs establish new local-to-global correspondences, and use methods of infinite-combinatorial topology, including a new fusion result of Francis Jordan.
Archive classification: math.GN math.CA math.CO math.FA math.LO
Remarks: Submitted for publication
Submitted from: tsaban@math.biu.ac.il
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1112.2373
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