1.

Define metric spaces with examples.

Answer»

Definitions

Let X be a set. A metric on X is an assignment of a distance d(x, y) ∈ R to every pair of "points" x, y in X 
(that is d: X × X→ R) satisfying the following:

  1. (Positivity) For all x, y ∈ X, d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y,

  2. (Symmetry) For all x, y ∈ X, d(x, y) = d(y, x),

  3. (The triangle inequality) For all x, y, z ∈ X, d(x, y) + d(y, z) ≥ d(x, z).


A metric space is a set X together with such a metric.

Examples

The prototype: The set of real numbers R with the metric d(x, y) = |x - y|.This is what is called the usual metric on R.


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