We provide a complete characterization of the solvability/impossibility of deterministic stabilizing consensus in virtually 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 show 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 continuous decision functions for terminating consensus, 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.