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On closed mappings of $ \sigma$-compact spaces and dimension


Authors: Elżbieta Pol and Roman Pol
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
Published electronically: June 10, 2019
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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

DOI: https://doi.org/10.1090/proc/14603
Keywords: Hilbert space, remainder, closed mapping, transfinite small inductive dimension, Effros Borel space
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
Article copyright: © Copyright 2019 American Mathematical Society