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：无许可分布式系统中，参与者无需中央权威授权即可自由加入或退出网络。现代许多分布式系统天然具有无许可特性——若设置中央许可权限机构反而会违背其设计初衷，这包括区块链、文件共享协议、部分投票系统等。由于无许可特性，此类系统具有异构性：参与者可能仅拥有系统的局部视图，且持有不同目标与信念。因此，传统意义上的共识（即系统级一致性）可能不再适用，我们需要对其进行泛化。这构成了一项挑战：我们应如何理解异构共识的本质？需要何种数学框架作为支撑？又如何基于此构建对鲁棒、高效且安全的无许可系统的理论认知与数学模型？本文采用半拓扑作为框架分析异构共识。半拓扑与拓扑类似，但取消了对开集交集仍为开集的限制。半拓扑理论体系丰富，与拓扑学密切相关，却发展出独特特征与数学结构。我们引入包括反豪斯多夫性质和"拓扑集"新概念在内的良性质条件，并阐明这些结构如何关联共识问题。我们将半拓扑限定为见证半拓扑——对应霍尔子句理论的可算法处理子类，具有优良数学性质。本文进一步提出并研究半拓扑特有的若干基本概念，揭示拓扑学中的经典量（如稠密集与闭包）在新的半拓扑语境下呈现的有趣且实用的新特性。