A distributed system is permissionless when participants can join and leave the network without permission from a central authority. Many modern distributed systems are naturally permissionless, in the sense that a central permissioning authority would defeat their design purpose: this includes blockchains, filesharing protocols, some voting systems, and more. By their permissionless nature, such systems are heterogeneous: participants may only have a partial view of the system, and they may also have different goals and beliefs. Thus, the traditional notion of consensus -- i.e. system-wide agreement -- may not be adequate, and we may need to generalise it. This is a challenge: how should we understand what heterogeneous consensus is; what mathematical framework might this require; and how can we use this to build understanding and mathematical models of robust, effective, and secure permissionless systems in practice? We analyse heterogeneous consensus using semitopology as a framework. This is like topology, but without the restriction that intersections of opens be open. Semitopologies have a rich theory which is related to topology, but with its own distinct character and mathematics. We introduce novel well-behavedness conditions, including an anti-Hausdorff property and a new notion of `topen set', and we show how these structures relate to consensus. We give a restriction of semitopologies to witness semitopologies, which are an algorithmically tractable subclass corresponding to Horn clause theories, having particularly good mathematical properties. We introduce and study several other basic notions that are specific and novel to semitopologies, and study how known quantities in topology, such as dense subsets and closures, display interesting and useful new behaviour in this new semitopological context.

翻译：无许可分布式系统允许参与者无需中央机构授权即可自由加入或退出网络。现代分布式系统（包括区块链、文件共享协议、部分投票系统等）天然具有无许可特性——引入中央授权机制反而会违背其设计初衷。这种无许可特性导致系统呈现异构性：参与者仅能观测系统局部信息，且可能持有不同目标与信念。因此，传统的系统级一致性共识概念可能不再适用，亟需对其进行泛化。这带来系列挑战：如何理解异构共识的本质？需要构建何种数学框架？又如何借助该框架建立对鲁棒、高效且安全的无许可系统的理论认知与数学模型？我们以半拓扑为框架分析异构共识。半拓扑类似拓扑，但取消了“开集交集仍为开集”的限制。半拓扑理论体系丰富，既与拓扑学存在深刻关联，又具有独特特征与独立数学结构。我们引入了包括反豪斯多夫性质与新型“开拓扑集”概念在内的良态条件，揭示这些结构与共识机制的内在联系。通过对半拓扑施加限制，得到见证半拓扑——对应Horn子句理论的可计算子类，兼具优越数学性质。我们提出并研究了半拓扑特有的多个基本概念，并深入探究稠密集、闭包等拓扑学经典量在此新型半拓扑语境中展现的独特且具有实际应用价值的新性质。