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On the collection of Baire class one functions on the irrationals


Author: Roman Pol
Journal: Proc. Amer. Math. Soc. 143 (2015), 4457-4462
MSC (2010): Primary 03E15, 26A21, 54H05
DOI: https://doi.org/10.1090/S0002-9939-2015-12583-X
Published electronically: March 25, 2015
MathSciNet review: 3373944
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Abstract: The Baire class one fan over the irrationals $ \mathbb{N} ^{\mathbb{N}}$ is the space $ S(\mathbb{N} ^{\mathbb{N}})$
$ = ( \mathbb{N} ^{\mathbb{N}} \times \mathbb{N})\cup \{ \infty \}$, where basic neighbourhoods of $ \infty $ are epigraphs of the first Baire class functions $ f: \mathbb{N}^{\mathbb{N}}\to \mathbb{N}$, augmented by $ \infty $, and the remaining points are isolated; $ S(\mathbb{N}) = ( \mathbb{N} \times \mathbb{N} )\cup \{\infty \}$ is the standard countable sequential fan. We prove that $ S(\mathbb{N}^{\mathbb{N}}) \times S(\mathbb{N})$ has countable tightness: in this product, whenever $ p \in \overline {A}$, then $ p\in \overline {B}$ for some countable $ B\subset A$.


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Additional Information

Roman Pol
Affiliation: Mathetmatics Institute, University of Warsaw, ul Banacha 2, PL 02-097, Warsaw, Poland
Email: R.Pol@mimuw.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-2015-12583-X
Keywords: Baire class one functions, tightness
Received by editor(s): March 23, 2014
Received by editor(s) in revised form: June 19, 2014
Published electronically: March 25, 2015
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2015 American Mathematical Society