Metric Spaces

A metric space is a set $X$ together with a function $$ \operatorname{dist}: X \times X \to \R $$ called a metric such that the following laws are satisfied:

1. (Positivity) $\operatorname{dist}(x,y) \ge 0$ with equality $x = y$
2. (Symmetric) $\operatorname{dist}(x,y) = \operatorname{dist}(y,x)$
3. (Triangle Inequality) $\operatorname{dist}(x,z) \le \operatorname{dist}(x,y) + \operatorname{dist}(y,z)$