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Topological aspects of hybrid processes (a treatment using non-standard analysis)

Hybrid systems are systems in which both physical and computational behavior play an im-portant role. In the study of such systems, using techniques from computer science, we oftenencounter problems of a topological nature. In this paper, we brieﬂy discuss three of theseproblems, namely: the continuity of physical behavior, the occurrence of Zeno-phenomenaand other limit-behavior due to the combination of computations and physical behavior, andthe inﬂuence of imprecise measurements.Before we are able to discuss the problems mentioned above, we need a way to statethem formally. For this purpose, we use topological transition systems, i.e. labeled transitionsystems hX,Σ,→i in which the state space X and signal space Σare both equipped with atopology (see e.g. [8]). Most of the work on hybrid systems that deals with Zeno-behavior,for example, implicitly assumes such a topology. But the consequences of that topology forequivalences, and the like, are hardly ever made precise (see [5, 15, 7] for some exceptions).To analyse these topological transition systems, we will use the methods of non-standardanalysis, also known as the mathematics of inﬁnitesimals. From a non-standard point ofview, the presence of a topology means that the spaces X and Σare lifted to spaces X ⊆