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ON GRAEV’S THEOREM FOR FREE PRODUCTS OF HAUSDORFF TOPOLOGICAL GROUPS

Part of: Structure and classification of infinite or finite groups Topological and differentiable algebraic systems General theory of categories and functors

Published online by Cambridge University Press:  29 March 2021

GURAM SAMSONADZE Open the ORCID record for GURAM SAMSONADZE [Opens in a new window]  and
DALI ZANGURASHVILI Open the ORCID record for DALI ZANGURASHVILI [Opens in a new window]
GURAM SAMSONADZE
Affiliation:
Georgian Technical University, 77 Kostava Str., Tbilisi0160, Georgia e-mail: g.samsonadze@gtu.ge
DALI ZANGURASHVILI*
Affiliation:
Andrea Razmadze Mathematical Institute, Tbilisi State University, 6 Tamarashvili Str., Tbilisi0177, Georgia
*
e-mail: dali.zangurashvili@tsu.ge
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Abstract

The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of $k_\omega $ -groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the $X_0$ -topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.

Keywords

free product of Hausdorff topological groupskω-groupLindelöf P-groupMal’cev initial topology

MSC classification

Primary: 22A05: Structure of general topological groups
Secondary: 18A30: Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 475 - 483
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author gratefully acknowledges the financial support from Shota Rustaveli Georgian National Science Foundation (Ref. FR-18-10849).

References

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