Pointwise convergence of partial functions: The Gerlits–Nagy Problem

For a set X⊆RX⊆R, let B(X)⊆RXB(X)⊆RX denote the space of Borel real-valued functions on XX, with the topology inherited from the Tychonoff product RXRX. Assume that for each countable A⊆B(X)A⊆B(X), each ff in the closure of AA is in the closure of AA under pointwise limits of sequences of partial functions. We show that in this case, B(X)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 XX.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.