Show that HX h is a metric space see Chapter 2 Definition 48

Show that (H(X), h) is a metric space, (see Chapter 2, Definition 4.8). DEFINITION 4.8 (The Hausdorff Distance). For compact sets A, B H(X) we let h(A, B) = max {d(A, B), d(B, A)}

Solution

problem 2) A metric space is a set, whose distance between its members are defined.

It is given in definition 4.8 that any A, B elements of H(x), the distance between A and B is defined and it is h(A,B). thus distance between every elements of the set is defined. so clearly (H(X), h) is a metric space.

Here distance between any 2 members is given as the maximum of distance between A,B or distance between B,A.