 Often one wishes to restrict attention to a class of Kripke models whose relations \(R_a\) satisfy certain desirable properties such as reflexivity, transitivity, Euclideanness, or seriality. Reflexivity tells us that agent knowledge is truthful, transitivity tells us that agents know what they know, Euclideanness tells us that agents know what they do not know, and seriality tells us that agent knowledge is consistent. (A belief reading is also possible.) In order to study public announcements over such classes, we must be certain that the public announcement of a formula F does not transform a given Kripke model M into a new model \(M[F!]\) that falls outside of the class. The following theorem indicates when it is that a given class of Kripke models is “closed” under public announcements (meaning a public announcement performed on a model in the class always yields another model in the class). 