By Stein M.R., Dennis R.K. (eds.)

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M. Ying, Topology in Process Calculus © Springer-Verlag New York, Inc. 2001 38 3. 1 Strong Limit Bisimulat ions In Ch apter 1, I pointed out that our approach to limits of agents is bas ed on the Moore-Smith theory of convergence in point-set topology. A key notion in this t heory is nets. The concept of nets is a generalization of sequ ences. As is well-known, sequences are able to characterize only first-countable topologies; by contrast nets are comp etent to accommodate any topology. We first recall the notion of nets and some other relat ed notions in point-set topology.

A ssume that for some sort H of P and fo r some sort K ofQ , H UK:I f . Th en P = Q if an d only if, f or all R , P +R;::::: Q+R. Proof. 2. Conversely, if P :I Q , t hen 1. there are a E A ct and pi E ~ such that P ~ pi and for all Q' , Q ~ Q' implies that P' ;::::: Q' does not hold; or 2. t he re ar e a E A ct and Q' E P such that Q ~ Q' and for all pi , P ~ pi implies that pi ;::::: Q' does not hold . We conside r only the first case; the second case is similar. Since P + R ;::::: Q + R for all RE p , we may t ake R = 0 , and then it hold s t hat P;::::: Q.

14. Let S S;; ~ x ~ and PSQ implies that, for all (i) whenever P ~ P' , then for som e Q' , Q ~ Q' and P' rv 0: E Act, S ;:::::: Q',. and (ii) whenever Q ~ Q', then for some P', P ~ P' and P' ;:::::: S rv Q'. Then S S;;;:::::: . Proof. Suppose that PSQ and P ~~ P' . 12 it suffices to show that there is Q' such that Q ~ Q' and P' ;:::::: S ;:::::: Q' . We proceed is exact ly the condition (i) . Let by induction on p. z, PI ~~ P'. Then from condition (i) we have Q ::f} Ql and PI rv P2SQ2 ;:::::: Ql for some P2, Ql and Q2' Furthermore, there exists P 2 such that P 2 ~~ P2 and P' rv P2.