Compactness in MV-Topologies: Tychonoff Theorem and Stone-Cech Compactification

TL;DR

This paper extends classical topological theorems to MV-topologies, establishing a Tychonoff theorem, exploring compactifications, and linking these results to foundational set theory axioms.

Contribution

It introduces a Tychonoff theorem for MV-topological spaces, extends Stone-Cech compactification, and connects these to the Axiom of Choice within fuzzy topology.

Findings

Proves a Tychonoff theorem for MV-topologies.

Establishes the existence of products and coproducts in relevant categories.

Shows the equivalence of the Tychonoff theorem to the Axiom of Choice in ZF.

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-Cech compactification functor to the category of MV-topological spaces, and we discuss its relationship with previous works on compactification for fuzzy topological spaces.