We are concerned with the question of whether a noncompact space with a nice local structure contains a ray, i.e., a closed homeomorph of $[0,1)$. We construct rays in incomplete locally path connected spaces, and also, in noncompact metrizable convex sets; as a consequence these spaces lack the fixed point property. On the other hand, we give an example of a noncompact (nonmetrizable) convex subset $C$ of a locally convex topological vector space $E$ which has the fixed point property.
"Rays and the fixed point property in noncompact spaces." Tsukuba J. Math. 21 (1) 97 - 112, June 1997. https://doi.org/10.21099/tkbjm/1496163163