Let $M$ be a manifold connected hausdorf noncompact. Then there is a closed embebdding of the half line $[0, \infty)$ into $M$.
I'm hard to build such a function without self-intersections.
Book: Differential Topology, Hirsch. p 27.
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An Idea: Let ${U_\alpha}{\alpha \in I}$ be an open cover without any finite subcover. Then, you intersect these open sets with your charts, and construct the embedding locally (i.e. if $V$ is a chart, you map $U\alpha \cap V$ to $W\subset \mathbb{R}^n$ and extend). – Kyle Gannon Feb 02 '16 at 03:11