Abstract
In this paper we study topologizability of structures. We extend the method of Kotov of topologizability of countable algebras to uncountable structures. We also show that in the case of topologizable relational countable structures the topology can be made metrizable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arnautov, V.: Nondiscrete topologizability of countable rings. Dokl. AN SSSR 191, 747–750 (1970). (Russian)
Banach, T., Protasov, I., Sipacheva, O.: Topologization of a set endowed with an action of a monoid. Topol. Appl. 169, 161–174 (2014)
Klyachko, A., Olshanskii, A., Osin, D.: On topologizable and non-topologizable groups. Topol. Appl. 160, 2104–2120 (2013)
Kotov, M.V.: On the topologizability of countable equationally Noetherian algebras. Algebra Log. 52, 105–115 (2013)
Markov, A.: On unconditionally closed sets. Mat. Sb. 18, 3–28 (1946). (Russian)
Olshanski, A.: A remark on a countable non-topologizable group. Vestn. Mosk. Gos. Univ. Mat. Mekh. 3, 103 (1980). (Russian)
Podewski, K.-P.: Topologisierung algebraischer Strukturen. Rev. Roum. Math. Pures Appl. 22(9), 1283–1290 (1977)
Shelah, S.: On a problem of Kurosh, Jonsson groups, and applications. In: Word Problems, II, Conf. on Decision Problems in Algebra, Oxford, 1976. Stud. Logic Found. Math., vol. 95, pp. 373–394. North-Holland, Amsterdam, New York (1980)
Sipacheva, O.: Unconditionally \(\tau \)-closed and \(\tau \)-algebraic sets in groups. Topol. Appl. 155, 335–341 (2008)
Sipacheva, O.: Consistent solution of Markov’s problem about algebraic sets, arxiv:math/0605558v2
Taimanov, A.D.: Topologization of countable algebras. Dokl. Akad. Nauk. SSSR 243, 284–286 (1978). (Russian)
Acknowledgements
The authors are grateful to the referee for very helpful remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by A. Dow.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Dutka, J., Ivanov, A. Topologizable structures and Zariski topology. Algebra Univers. 79, 72 (2018). https://doi.org/10.1007/s00012-018-0557-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-018-0557-1