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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Free products of topological groups which are $ k\sb{\omega }$-spaces


Author: Edward T. Ordman
Journal: Trans. Amer. Math. Soc. 191 (1974), 61-73
MSC: Primary 22A05
MathSciNet review: 0352320
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Abstract: Let G and H be topological groups and $ G \ast H$ their free product topologized in the manner due to Graev. The topological space $ G \ast H$ is studied, largely by means of its compact subsets. It is established that if G and H are $ {k_\omega }$-spaces (respectively: countable CW-complexes) then so is $ G \ast H$. These results extend to countably infinite free products. If G and H are $ {k_\omega }$-spaces, $ G \ast H$ is neither locally compact nor metrizable, provided G is nondiscrete and H is nontrivial. Incomplete results are obtained about the fundamental group $ \pi (G \ast H)$. If $ {G_1}$ and $ {H_1}$ are quotients (continuous open homomorphic images) of G and H, then $ {G_1} \ast {H_1}$ is a quotient of $ G \ast H$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0352320-6
PII: S 0002-9947(1974)0352320-6
Keywords: Free product of topological groups, k-space, $ {k_\omega }$-space, CW-complex, quotient of topological groups, nonmetrizable group
Article copyright: © Copyright 1974 American Mathematical Society