The direct method of Liapunov characterizes stability properties of sets in dynamical systems in terms of the existence of corresponding real-valued “Liapunov functions”. The traditional limitation of Liapunov functions to real values has blocked the extension of this approach to more general systems. In this paper, a stability concept analogous to classical uniform stability is defined as a relationship between a quasiorder and a uniformity on the same set, and it is shown that stability in this sense occurs if and only if there exists a Liapunov function taking values in a certain partially ordered uniform space associated with the given uniformity and called its retracted scale. A few general properties of scales and retracted scales are discussed, and the continuity of the Liapunov functions is briefly considered.
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