| Résume | We consider expansions M of the real line (R,<) having the property that, for all sets E definable in M, each connected component of M is definable in M; we then say that M is "component closed". Some notable examples are: (a) o-minimal M; (b) M=(R,<,+,x,Z); and (c) the "component closure" of M (defined in an obvious way). I will demonstrate that, in contrast to cases (a) and (b), the question "Is M component closed?" can be difficult to answer even if the model theory of M is well understood. This is very preliminary joint work with Athipat Thamrongthanyalak. |
