The set of all eventually periodic words is not Buchi Recognizable?
Note that every non-empty Büchi recognizable language contains at least one eventually periodic word and that the class of all Büchi recognizable languages is closed under complement. Thus, if the language $L = \{w \in \Sigma^\omega \mid w \text{ is eventually periodic}\}$ was Büchi recognizable its complement $L^C$ is also Büchi recognizable but of course contains no eventually periodic word contradicting the first fact I stated.