(Solved): Consider a subset X of Rn. (a) If X is unbounded, show that F={Br(0)r=1,2,3,} is a cover o ...

Consider a subset $X$ of ${\mathbb{R}}^n$. (a) If $X$ is unbounded, show that $\mathcal{F}=\left\{B_r(0)?r=1,2,3,\dots \right\}$ is a cover of $X$ that does not have a finite subcover. (b) If $X$ is not closed, let $y$ \& $X$ be a point such that every ball $B_{?}(y)$ with $?>0$ intersects $X$. Show that $\mathcal{F}=\{{\mathbb{R}}^n?B_{1/r}[y]?r=1,2,3,\dots$.$?isanopencoverof$ $X$ that does not have a finite subcover.