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.

Orenshtein, T., Tsaban, B. (2013). Pointwise convergence of partial functions: The Gerlits-Nagy Problem. ADVANCES IN MATHEMATICS, 232(1), 311-326 [10.1016/j.aim.2012.09.017].

Pointwise convergence of partial functions: The Gerlits-Nagy Problem

Orenshtein T.
;
2013

Abstract

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.
Articolo in rivista - Articolo scientifico
Baire class 1 functions; Borel covers; Covering properties; Fréchet-Urysohn spaces; Gerlits-Nagy Problem; Pointwise convergence; Selection principles; Sequential closure; Sequential spaces;
English
16-ott-2012
2013
232
1
311
326
none
Orenshtein, T., Tsaban, B. (2013). Pointwise convergence of partial functions: The Gerlits-Nagy Problem. ADVANCES IN MATHEMATICS, 232(1), 311-326 [10.1016/j.aim.2012.09.017].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/362302
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