First a bit of background: I have a particular $C^1$ codimension one foliation $\mathscr{F}$ of $P$ (n-dimensional Euclidean space with all positive coordinates) which I have equipped with a total order on the leaves. I know that the leaf space $\mathscr{L}$ is a one-dimensional manifold and, because of the type of foliation, I also know that the leaf space is connected and Hausdorff. I'd like to be able to demonstrate that the canonical projection onto $\mathscr{L}$ can be treated as if it is a map into $R_+$ (strictly positive real line) which respects the order on the leaves.
So I've been researching this and came across the following: In "Geometric Theory of Foliations" by Camacho and Lins Neto there is the statement "...on a simply connected, one-dimensional manifold [I'll call it $M$] (Hausdorff or not) there always exists a continuous, strictly monotonic function". Fine. My question is: Doesn't monotonicity require that there be an order on the domain? If so then what order is on $M$? My suspicion is that we know (by classification of 1-manifolds) that there is a homeomorphism of $\mathscr{L}$ and $R_+$, and we can get an order on each local chart, but it's not global. Is there a result that says if a map is locally monotonic then it's globally so?
I just want to make sure I'm thinking correctly, thank you.