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.

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