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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On closed mappings of $\sigma$-compact spaces and dimension
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by Elżbieta Pol and Roman Pol PDF
Proc. Amer. Math. Soc. 147 (2019), 5009-5017 Request permission

Abstract:

A remainder of the Hilbert space $l_{2}$ is a space homeomorphic to $Z \setminus l_{2}$, where $Z$ is a metrizable compact extension of $l_{2}$, with $l_{2}$ dense in $Z$. We prove that for any remainder $K$ of $l_{2}$, every non-one-point closed image of $K$ contains either a compact set with no transfinite dimension or compact sets of arbitrarily high inductive transfinite dimension $\mathrm {ind}$. We shall also construct for each natural $n$ a $\sigma$-compact metrizable $n$-dimensional space whose image under any non-constant closed map has dimension at least $n$ and analogous examples for the transfinite inductive dimension $\mathrm {ind}$ (this provides a rather strong negative solution of a problem in [Dissertationes Math. (Rozprawy Mat.) 216 (1983), pp. 1–41]).
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Additional Information
  • Elżbieta Pol
  • Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
  • Email: E.Pol@mimuw.edu.pl
  • Roman Pol
  • Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
  • Email: R.Pol@mimuw.edu.pl
  • Received by editor(s): February 12, 2018
  • Received by editor(s) in revised form: February 24, 2019
  • Published electronically: June 10, 2019
  • Communicated by: Ken Bromberg
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 5009-5017
  • MSC (2010): Primary 54E40, 54F45, 57N20; Secondary 54D40, 54H05
  • DOI: https://doi.org/10.1090/proc/14603
  • MathSciNet review: 4011532