0.2 Metic Spaces

Question 1

What is a metric space?

Answer 1

A metric space is a set X, possessing a function d such that:

d: X x X —> R
satisfying the following axioms:
1. d(x, y) >= 0
2. d(x, y) = 0 iff x = y
3. d(x, y) = d(y, x)
4. d(x, y) <= d(x, z) + d(z, y)

Question 2

What are 4 examples of metric spaces?

Answer 2

1. The usual metric on R
2. The usual metric on R^n
3. The metric dp for p >= 1
4. The metric d infinity