(a) $d(x,A)=0$ iff for any $\epsilon>0$ : $B(x,\epsilon)\cap A\neq\emptyset$ iff $x\in\overline{A}$ .(b) $A$ is compact, $d$ is continuous on $X\times X$ and, therefore, continuous in both variables, hence, for a given $x$ : $d(x,a)$ reaches the minimum on $A$ .(c) $x\in U(A,r)$ iff $d(x,a)<r$ for some $a\in A$ iff $x\in\cup_{a\in A}B(a,r)$ .(d) For $a_j\in A$ let $r_j$ be such that $B(a_j,2r_j)\subseteq U$ . Balls $B(a_j,r_j)$ cover $A$ , there is a finite subcovering $a_1, . . ., a_n$ , let $\epsilon$ be the minimum of corresponding $r_i$ ’s. For $y\in B(a_i,r_i)$ : $B(y,\epsilon)\subseteq B(a_i,r_i+\epsilon)\subseteq B(a_i,2r_i)\subseteq U$ .(e) The idea is that there is no finite subcovering and $\epsilon$ as the minimum of $r_j$ ’s must be zero. For example, $x\times 1/x$ for $x>0$ has no $\epsilon$ -neighborhood in $\mathbb{R}_+\times\mathbb{R}_+$ .