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 $R$?