Metric

MT#DEF. Metric.

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

1. Separation. For all $x$, $y\in X$,

   \[d(x,y) = 0 \enspace\lrimp\enspace x=y.\]

2. Symmetry. For all $x$, $y\in X$,

   \[d(x,y) = d(y,x).\]

3. 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.

1. The empty function $\varnothing$ is a metric on $\varnothing$ (the empty metric).

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

MT#PROP-POS. 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.

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

MT#PROP-SUB. Metric Subspace, Induced Metric.

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 $X$.