Metric space and topology.

1. Consider the metric space $(X = \Z, d_X(x,y) = |x-y|)$. Write down the open ball $B_3 (2)$, $B_{1/2}(2)$. Is the subset $\{2\}$ open in $X$? Is it closed in $X$? Explain.

2. Consider the metric space $(X = [0,1) \In \R, d_X(x,y) = |x-y|)$. Write down the open ball $B_{3} (0)$, $B_{1/3}(0)$. Is the subset $[0,1)$ open in $X$? closed in $X$? Is the subset $[1/3, 1)$ open in $X$? closed in $X$?

1. Let $(X, d)$ be a metric space. Let $Conf_2(X) = \{ S \In X, |S|=2\}$, i.e., an element in $Conf_2(X)$ is subset $S \In X$ consisting of two points. For example, let $X = \R$, $S = \{ 2, -1.1 \} \In X$, then $S$ is an element in the set $Conf_2(X)$. Can you put a metric on $Conf_2(X)$?