When we work in \(\mathbb {R}\) with the usual metric, we think of distance as measured by absolute value. Points are close when the absolute value of the difference is small. We might reasonably argue that points x and y are close when they satisfy \(|y - x| < r\) , where r is a small positive number; that is to say, y is in the open interval \((x - r, x + r)\) . This interpretation allows us to visualize the distance between the points. As we will see in this chapter, all metrics have this visual interpretation.