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

Part of: Topological and differentiable algebraic systems Special properties

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 [email protected]
Arkady Leiderman
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva, Israel email [email protected]
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 [email protected]
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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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