The Hegselmann-Krause (HK) model is a typical self-organizing system with local rule dynamics. In spite of its widespread use and numerous extensions, the underlying theory of its synchronization induced by noise still needs to be developed. In its original formulation, as a model first proposed to address opinion dynamics, its state-space was assumed to be bounded, and the theoretical analysis of noise-induced synchronization for this particular situation has been well established. However, when system states are allowed to exist in an unbounded space, mathematical difficulties arise, whose theoretical analysis becomes non-trivial and is as such still lacking. In this paper, we completely resolve this problem by exploring the topological properties of HK dynamics and employing the theory of independent stopping time. The associated result in full state-space provides a solid interpretation of the randomness-induced synchronization of self-organizing systems.