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EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP $A(X\oplus X)$ INTO $A(X)$

Part of: Special properties Topological and differentiable algebraic systems

Published online by Cambridge University Press:  17 April 2019

Mikołaj Krupski ,
Arkady Leiderman  and
Sidney Morris
Mikołaj Krupski
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland email mkrupski@mimuw.edu.pl
Arkady Leiderman
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva, Israel email arkady@math.bgu.ac.il
Sidney Morris
Affiliation:
School of Science, Engineering and Information Technology, Federation University Australia, PO Box 663, Ballarat, Victoria, 3353, Australia Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia email morris.sidney@gmail.com
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Abstract

We consider the following question: for which metrizable separable spaces $X$ does the free abelian topological group $A(X\oplus X)$ isomorphically embed into $A(X)$. While for many natural spaces $X$ such an embedding exists, our main result shows that if $X$ is a Cook continuum or $X$ is a rigid Bernstein set, then $A(X\oplus X)$ does not embed into $A(X)$ as a topological subgroup. The analogous statement is true for the free boolean group $B(X)$.

MSC classification

Primary: 22A05: Structure of general topological groups 54F15: Continua and generalizations
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 708 - 718
Copyright
Copyright © University College London 2019 

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