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EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP A ( X ⊕ X ) INTO A ( X )
Author(s) -
Krupski Mikołaj,
Leiderman Arkady,
Morris Sidney
Publication year - 2019
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579319000123
Subject(s) - mathematics , embedding , abelian group , topological group , separable space , group (periodic table) , metrization theorem , free group , topological ring , g module , pure mathematics , topology (electrical circuits) , topological space , discrete mathematics , elementary abelian group , combinatorics , topological vector space , mathematical analysis , chemistry , organic chemistry , artificial intelligence , computer science
We consider the following question: for which metrizable separable spaces X does the free abelian topological group A ( X ⊕ 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 ⊕ X ) does not embed into A ( X ) as a topological subgroup. The analogous statement is true for the free boolean group B ( X ) .