Metric Spaces

A metric space is a very simple and general mathematical space which consists of a set, \(M\), and a binary function, \(d(x, y)\), called the distance or metric function which gives us the ability to compute lengths on the set \(M\).

The distance function is defined as,

\[ d: M \times M \rightarrow 0 \cup \mathbb{N}\,, \]

and needs to satisfy several properties.

  1. \(d(x, y) \geq 0\). The distance function \(d(x, y)\) cannot be negative.

  2. \(d(x, y) = 0\) if and only if \(x = y\). Therefore, the result is only zero if we are evaluating the distance from a point to itself.

  3. \(d(x, y) = d(y, x)\). The distance should be the same regardless of how we evaluate it, from \(x\) to \(y\) or from \(y\) to \(x\).

  4. \(d(x, y) \leq d(x, z) + d(z, y)\). The triangle inequality must hold. If we have three segments, \(x\) to \(y\), \(x\) to \(z\), and \(y\) to \(z\), then the sum of two of the segments will always be larger than the other one. This follows from the Pythagorean theorem.