Given an open subset $I \subset \mathbb{R}$ if $(x_1-r_1,x_1+r_1) \subset I$ does that imply that $[x_1-r_1,x_1+r_1] \subset I$? If not, could I choose $r_1>0$ that such that this holds?
More generally, Let $X$ be a complete metric space. If $W$ is open and $B(x_1,r_1) \subset W$, does it imply that $B[x_1,r_1] \subset W$? Could I choose such a $r_1$?
Thanks.