Semitopology: a topological approach to decentralized collaborative action

Abstract We introduce semitopology, a generalization of point-set topology that removes the restriction that intersections of open sets need necessarily be open. The intuition is that points represent participants in a decentralized system, and open sets represent collections of participants that collectively have the authority to collaborate to update their local state; we call this an actionable coalition. Examples of actionable coalition include: majority stakes in proof-of-stake blockchains; communicating peers in peer-to-peer networks; and even pedestrians working together to not bump into one another in the street. Where actionable coalitions exist, they have in common that collaborations are local (updating the states of the participants in the coalition, but not immediately those of the whole system); collaborations are voluntary (up to and including breaking rules); participants may be heterogeneous in their computing power or in their goals (not all pedestrians want to go to the same place); participants can choose with whom to collaborate; and they are not assumed subject to permission or synchronization by a central authority. We develop a topology-flavoured mathematics that goes some way to explaining how and why these complex decentralized systems can exhibit order, and gives us new ways to understand existing practical implementations. Semitopology is also interesting in and of itself, having a rich and interesting theory that quickly deviates from standard accounts on topological spaces. It soon becomes clear that the most interesting semitopologies are rather ill-behaved from the usual viewpoint, as they are never Hausdorff. A notion of ‘transitive open sets’ (topens) becomes central to the story, as topens define subsets of participants who should decide the same value in a distributed system that tries to achieve consensus, and points are called ‘regular’ when they have a topen neighbourhood. The theory is then further developed by introducing intertwined points, closures, closed sets and two interesting characterizations of regularity.