# 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.