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Compactness in MV-topologies: Tychonoff theorem and Stone–Čech compactification

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Abstract

In this paper, we discuss some questions about compactness in MV-topological spaces. More precisely, we first present a Tychonoff theorem for such a class of fuzzy topological spaces and some consequence of this result, among which, for example, the existence of products in the category of Stone MV-spaces and, consequently, of coproducts in the one of limit cut complete MV-algebras. Then we show that our Tychonoff theorem is equivalent, in ZF, to the Axiom of Choice, classical Tychonoff theorem, and Lowen’s analogous result for lattice-valued fuzzy topology. Last, we show an extension of the Stone–Čech compactification functor to the category of MV-topological spaces, and we discuss its relationship with previous works on compactification for fuzzy topological spaces.

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Notes

  1. What we call strong compactness here is called fuzzy compactness in the theory of lattice-valued fuzzy topologies [4]. It is worth remarking, however, that such a concept appears a very few times in the literature, since it is too much restrictive. Indeed, for example, even a fuzzy topological space with finite support may not be strongly compact.

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Funding

The funding was provided by Fapesb (Grant No. APP0072/2016) and Colciencias (Grant No. Ph.D. scholarship doctoral scholarship “Doctorado Nacional-567”).

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  1. Departamento de Matemáticas, Universidad del Valle, Cali, Colombia

    Luz Victoria De La Pava

  2. Departamento de Matemática, Universidade Federal da Bahia, Salvador, Bahia, Brazil

    Ciro Russo

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  1. Luz Victoria De La Pava
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  2. Ciro Russo
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Correspondence to Ciro Russo.

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De La Pava, L.V., Russo, C. Compactness in MV-topologies: Tychonoff theorem and Stone–Čech compactification. Arch. Math. Logic 59, 57–79 (2020). https://doi.org/10.1007/s00153-019-00679-6

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