Pointwise convergence of partial functions: The Gerlits-Nagy Problem

For a set ⊆R, let B(X)⊆RX denote the space of Borel real-valued functions on X, with the topology inherited from the Tychonoff product RX. Assume that for each countable A ⊆ 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 Fréchet-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 Hernańdez, we show that the same result holds for the space of all Baire class 1 functions on X.We conjecture that, in the general context, the answer to the continuous version of this problem is negative, but we identify a nontrivial context where the problem has a positive solution.The proofs establish new local-to-global correspondences, and use methods of infinite-combinatorial topology, including a new fusion result of Francis Jordan. © 2012 Elsevier Ltd.