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Are finite affine topological systems worthy of study?

  • Fuzzy systems and their mathematics
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Abstract

There exists the notion of topological system of S. Vickers, which provides a common framework for both topological spaces and the underlying algebraic structures of their topologies—locales. A well-known result of S. A. Morris states that every topological space is homeomorphic to a subspace of a product of a finite (three-element) topological space. We have already shown that the space of S. A. Morris is (in general) no longer finite in case of affine topological spaces (inspired by the concept of affine set of Y. Diers), which include many-valued topology. This paper provides an analogue of the result of S. A. Morris for topological systems of S. Vickers, and also shows that for affine topological systems, an analogue of the above three-element space becomes (in general) infinite. A simple message here is that finite systems play a (probably) less important role in the affine topological setting (for example, in many-valued topology) than they do play in the classical topology.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, 44242, Ohio, USA

    Jeffrey T. Denniston

  2. Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague, Kamýcká 129, 16500, Prague, Czech Republic

    Sergey A. Solovyov

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  1. Jeffrey T. Denniston
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  2. Sergey A. Solovyov
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Denniston, J.T., Solovyov, S.A. Are finite affine topological systems worthy of study?. Soft Comput 26, 9021–9033 (2022). https://doi.org/10.1007/s00500-022-07260-z

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