Metric

Definition 1. Let $X$ be a non-empty set. A function $d : X \times X \to \R$ is a metric on $X$ if:

(Separation) For all $x$, $y \in X$,
\[d(x, y) = 0 \enspace\lrimp\enspace x = y.\]
(Symmetry) For all $x$, $y \in X$,
\[d(x, y) = d(y, x).\]
(Triangle Inequality) For all $x$, $y$, $z \in X$,
\[d(x, z) \leq d(x, y) + d(y, z).\]

$(X, d)$ is then called a metric space.

Examples.

The empty function $\empt$ is a metric on $\empt$ (the empty metric).
The absolute difference $d : (x, y) \mapsto \lvert x - y \rvert$ is a metric on $\R$ (the standard metric).

Definition 2. Let $(X, d)$ be a metric space and $Y \subseteq X$. Then $d \restriction_Y$ is a metric on $Y$, called an induced metric, and $(Y, d \restriction_Y)$ is called a metric subspace of $X$.

Proposition 1 (Positivity). Let $(X, d)$ be a metric space. For all $x$, $y \in X$:

\[x \neq y \enspace\rimp\enspace d(x, y) > 0.\]

Proof.

\[2 d(x, y) = d(x, y) + d(x, y) = d(x, y) + d(y, x) \geq d(x, x) = 0.\]

If $x \neq y$, then $d(x, y) \neq 0$.