This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multimodal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in Fagin et al [4]), but we reify aspects of the machinery in the logical language.Special cases of our logic have been considered in Plaza [13], Gerbrandy [5,6], and Gerbrandy and Groeneveld [7]. The latter group of papers introduce a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions, and by using a more general semantics.Our logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power.

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Abstract. The aim of the present work is to develop a dualization of the Fraïssé limit construction from model theory and to indicate its surprising connections with the pseudo-arc. As corollaries of general results on the dual Fraïssé limits, we obtain Mioduszewski's theorem on surjective universality of the pseudo-arc among chainable continua and a theorem on projective homogeneity of the pseudo-arc (which generalizes a result of Lewis and Smith on density of homeomorphisms of the pseudo-arc among surjective continuous maps from the pseudo-arc to itself). We also get a new characterization of the pseudo-arc via the projective homogeneity property.

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We prove that arbitrary homomorphisms from one of the groups Homeo(2 N ), Homeo(2 N ) N , Aut(Q, <), Homeo(R), or Homeo(S 1 ) into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result on V.G. Pestov, that any action of the discrete group Homeo+(R) by homeomorphisms on a compact metric space has a fixed point.