Well-known metric spaces for modelling finitely branching and image finite systems are shown to be (the carrier of) terminal coalgebras. Introduction In the area of metric semantics, various metric structures have been proposed to model a wide spectrum of programming notions (see, e.g., [BV96]). In this paper, we focus on metric structures for modelling nondeterministic systems which may give rise to both terminating and nonterminating computations. The systems we have in mind are labelled transition systems [Kel76]. A large variety of programming notions can be modelled by means of these systems (see, e.g., [Plo81]). The models we consider are linear (cf. [Pnu85]). In these models, the locations in a computation where a nondeterministic choice is made are not visible. These linear models are usually contrasted with branching models (cf. [Gla90]). In those models, the positions in the computation where a nondeterministic choice is made are administrated. Typical examples of linear me...

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