Abstract
In this paper, we examine topological spaces that can be effectively defined over factor sets modulo equivalences on the set of natural numbers. We formulate a criterion of computable (effective) separability of topological spaces in terms of the approximability of the corresponding algebras by negative (uniformly effectively separated) algebras. We compare negative and positive algebra representations from the standpoint of the structure of the corresponding effective spaces. For effective infinite topological spaces, we prove the existence of their infinite effective compact extensions.
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Dedicated to the memory of Academician T. N. Kary-Niyazov
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 144, Proceedings of the Conference “Problems of Modern Topology and Its Applications” (May 11–12, 2017), Tashkent, Uzbekistan, 2018.
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Kasymov, N.K., Khodzhamuratova, I.A. Topological Spaces Over Algorithmic Representations of Universal Algebras. J Math Sci 245, 311–322 (2020). https://doi.org/10.1007/s10958-020-04692-6
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DOI: https://doi.org/10.1007/s10958-020-04692-6