Let d be a metric on X. Determine all constants k such that (i) kd,(ii) d + k is a metric on X.

Find all metrics on a set X consisting of two points. Consisting of one point.

Show that d(x, y) = √ |x-y|   defines a metric on the set of all real
numbers.

Does d (x, y) = (x - y)2 define a metric on the set of all real numbers?

Show that the real line is a metric space.

Show that the real line is a metric space.