Functorial relationships between lattice-valued topology and topological systems

Abstract

This paper investigates functorial relationships between lattice-valued topology (arising from fuzzy sets and fuzzy logic) and topological systems (arising from topological and localic aspects of domains and finite observational logic in computer science). Two such relationships are embeddings from TopSys into Loc-Top, both having two fold significance: for computer science the significance is that TopSys is not topological over Set × Loc, yet Loc-Top is topological over Set × Loc; hence these embeddings can be used to construct in Loc-Top the unique initial [final] lifts of all forgetful functor structured sources [sinks] in TopSys; and for topology, the significance is that both embeddings generate anti-stratified topological spaces from ordinary topological spaces and spatial locales rewritten as topological systems, thus justifying the current structural axioms of Loc-Top and lattice-valued topology (which include all anti-stratified, non-stratified, and stratified spaces).