Page:Topological Groups-Deane Montgomery.djvu/9

Rh distance from x to y, which satisfies the following axioms:


 * 1) d(x, y) = 0 if and only if x = y;


 * 2) d(x, y) = d(y, x);


 * 3) d(x, y) + d(y, z) ≥ d(x, z).

Any subset of a metric space is also a metric space. Ordinary three-space, or any subset of it, is a metric space. The following are other examples:

(1) n-dimensional space. This is the set of all sequences of n real numbers with distance (the metric) defined = d(x, y) where x = (x₁, in the space. 2 ...9 = · [(x₁ -√₂ ) ² + .... + (x₂) x) and y = (y₁, a(f, g) = .... (2) The set of all functions which are continuous on the closed interval [0, 1], with distance defined by -√₂ ) ² ¹/², 0< x≤ 1 y are two points max |f(x) = g(x)\. - (3) If R₁ and R₂ are metric spaces, their product is defined to be the set of all pairs (x₁, x₂) with distance given by 2 a[(x₁, x₂), (√₁, ³₂)] = { [ª₁(x₁, x₂)] ² + [ª₂(x₂, ₂)] 21/2