Pointwise convergence of partial functions: The Gerlits-Nagy Problem

TL;DR

This paper proves that under certain pointwise convergence conditions, the space of Borel functions is countably Fréchet–Urysohn, solving a longstanding problem and extending results to Baire class 1 functions, with implications in infinite combinatorial topology.

Contribution

It establishes the countable Fréchet–Urysohn property for Borel and Baire class 1 function spaces under specific convergence assumptions, solving a problem of Arnold Miller.

Findings

B(X) is countably Fréchet–Urysohn under pointwise convergence conditions

The result extends to Baire class 1 functions on X

New local-to-global methods and a fusion technique are developed

Abstract

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ 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 $A\sbst B(X)$, 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 Fr\'echet--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 Herna\'ndez, 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.