Point-free MV-topologies

TL;DR

This paper develops a point-free framework for MV-topologies, extending classical and fuzzy topology, and establishes dualities between algebraic and topological structures.

Contribution

It introduces new frame-like structures for MV-topologies and demonstrates dualities and sobriety characterizations within this point-free setting.

Findings

Established an adjoint relationship between fuzzy topological structures and frame-like structures.

Extended the class of MV-topological spaces with new point-free structures.

Characterized sobriety using neighborhood systems in the new framework.

Abstract

We propose a point-free approach to MV-topological spaces in the wake of previous works on both classical and fuzzy topology. In order to do that, we introduce suitable frame-type structures and a class of fuzzy topological spaces which includes and suitably extends the one of MV-topological spaces. Then we show an adjoint situation between such structures, and restrict such an adjointness to a duality between the corresponding classes of ``spatial frames'' and ``sober spaces''. We also use neighbourhood systems to characterize sobriety in this context.