Given a separable metric space $X$, what are some ways of forming a new metric space $Y$ such that:
(i) $Y$ contains a ray $R\simeq [0,\infty)$ ($\simeq$ means homeomorphic);
(ii) $R$ is open and dense in $Y$;
(iii) $X\simeq Y\setminus R$.
Moreover, can any non-compact separable metric space play the role of $R$?