On behavioural pseudometrics and closure ordinals

A behavioural pseudometric is often defined as the least fixed point of a monotone function F on a complete lattice of 1-bounded pseudometrics. According to Tarski@?s fixed point theorem, this least fixed point can be obtained by (possibly transfinite) iteration of F, starting from the least element @? of the lattice. The smallest ordinal @a such that F^@a(@?)=F^@a^+^1(@?) is known as the closure ordinal of F. We prove that if F is also continuous with respect to the sup-norm, then its closure ordinal is @w. We also show that our result gives rise to simpler and modular proofs that the closure ordinal is @w.

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