We provide a complete characterization of the solvability/impossibility of deterministic stabilizing consensus in any computing model with benign process and communication faults using point-set topology. Relying on the topologies for infinite executions introduced by Nowak, Schmid and Winkler (JACM, 2024) for terminating consensus, we prove that semi-open decision sets and semi-continuous decision functions as introduced by Levin (AMM, 1963) are the appropriate means for this characterization: Unlike the decision functions for terminating consensus, which are continuous, semi-continuous functions do not require the inverse image of an open set to be open and hence allow to map a connected space to a disconnected one. We also show that multi-valued stabilizing consensus with weak and strong validity are equivalent, as is the case for terminating consensus. By applying our results to (variants of) all the known possibilities/impossibilities for stabilizing consensus, we easily provide a topological explanation of these results.

翻译：我们利用点集拓扑学，对任何具有良性进程和通信故障的计算模型中确定性稳定共识的可解性/不可解性给出了完整的特征刻画。基于Nowak、Schmid和Winkler（JACM，2024）为终止共识引入的无限执行拓扑，我们证明了Levin（AMM，1963）提出的半开决策集与半连续决策函数是进行该特征刻画的恰当工具：与终止共识中使用的连续决策函数不同，半连续函数不要求开集的原像为开集，因而能够将连通空间映射到非连通空间。我们还证明了具有弱有效性与强有效性的多值稳定共识是等价的，这与终止共识的情况一致。通过将我们的结果应用于所有已知的稳定共识可能性/不可能性（及其变体），我们为这些结果提供了简洁的拓扑学解释。