According to the book *Infinite homotopy theory* by Baues-Quintero, if $X$ is a locally compact, locally connected, connected metrizable space, its Freudenthal ends can be identified (Proposition 9.20) by certain equivalence classes of rays. 


In particular, if $X$ is non-compact, there is a proper map $r:[0,\infty) \rightarrow X$.   
 1) Can $r$ be chosen so that it is also an embedding?  
 2) If yes, does $X$ have a metric for which $r$ can be chosen as an isometric embedding?



Rays in non-compact spaces
The answer to both questions is yes.


For the first question, recall that if $W$ and $X$ are non-compact
locally compact
Hausdorff spaces, a mapping $f: W \to X$ is continuous and proper iff
the mapping between their respective one point compactifications 
$\overline{f}: W \cup \{\infty\} \to X \cup \{\infty\}$ with 
$\overline{f}(\infty) = \infty$ and $\overline{f}|\_W = f$ is continuous.
In particular, if $W = [0,\infty]$ there is a path in $X\cup\{\infty\}$
from $f(0)$ to $\infty$. The image of this path is a Peano space and
therefore arc-connected, so there is an arc from $f(0)$ to $\infty$.
After appropriate re-parametrisation and restriction this gives us a
proper continuous injection $r: [0, \infty) \to X$. Since a proper
continuous mapping between locally compact Hausdorff spaces is closed,
this is a homeomorphic embedding.


The fact that the embedding is closed also provides the answer to the second question, as any compatible metric on a closed subspace of a
metrizable space can be extended to the whole space. If desired,
the extension can be complete, as described in


Bacon, P. (1968). *[Extending a Complete Metric.](http://dx.doi.org/10.2307/2313788)* The American Mathematical Monthly, 75(6), 642-643.


