It seems like something that could (should?) be true, but with topology you never know (unless you prove it...). EDIT: I'll be more exact: let [itex](X,\mathcal T)[/itex] be a topological space with X a totally ordered set and [itex]\mathcal T[/itex] the order topology. Say X is connected and [itex]A \subset X[/itex] is convex (i.e. [itex]\forall a,b \in A: a < b \Rightarrow [a,b] \subset A[/itex]). Is A connected?