(Solved): 2. (a) Let \( (X, d) \) be a metric space, \( B \subseteq X \) and \( x \in X \). A point \( a \in ...

2. (a) Let \( (X, d) \) be a metric space, \( B \subseteq X \) and \( x \in X \). A point \( a \in B \) is called a nearest point of \( B \) to \( x \) if \( d(x, a)=\operatorname{dist}(x, B) \). i) Show that if \( x \in B \) then \( x \) is the only nearest point of \( B \) to \( x \). ii) Show that if \( (X, d)=\left(\mathbb{R}, d_{\text {mas and }}\right) \) then there are atmost two mearest points of \( B \) to \( x \in \mathbb{R} \). Give an example to illustrate each of the following cases: 1. A case where there is no nearest point of \( B \) to \( x \). II. A case where there is one nearest point of \( B \) to \( x \). III. A case where there is two nearest point of \( B \) to \( x \). (b) Show that if \( (X, d) \) is a metric space and \( x \neq y \in X \) then there is \( r>0 \) such that \( B_{r}(x) \cap B_{r}(y)=\emptyset . \)