Generalizations of Topology with an Eye on Stone Duality
by
M. Andrew Moshier
Chapman University

One view of point-set topology is that it is the study of concrete representations of frames, i.e., complete lattices in which finite meet distributes over arbitrary join. The utility of this view comes into clear focus when we consider Stone Duality and its close relatives such as Priestley Duality. Recently, the author jointly with Achim Jung has found a bitopological setting in which Stone and Priestley are unified, but this unification leaves open the question of how to relate topology (as the study of concrete representations of frames) to bitopology (as the study of concrete representations of what?). Similarly, "fuzzy topology", where the non-classical logic of truth-values in [0,1] underpins the definition, can be understood as another generalization of topology. But this also leaves open questions of how to understand fuzzy topology in terms of concrete representations, particularly in light of the ideas surrounding Stone duality.

In this paper, we present a generalization of topological duality theorems in which topology, bitopology and fuzzy topology appear as special cases. The key idea is to reconsider the axioms for frames in light of Belnap's observation that logical and informational content of a proposition are distinct notions. In this light, frames can be seen as conflating positive logic (distributive lattices) and information (completeness). By treating logic and information as distinct notions, we obtain a useful category of 'truth-value objects' from which topology, bitopology and fuzzy topology are obtained as concrete categories by varying the truth-value object. Time permitting, we will also consider other examples, particularly, one in which entangled states of a qubit provide the``truth-values.''

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Date received: May 14, 2007