Pseudometric

Pseudometrics are metrics that do not seperate points. There can be two points with zero distance to each other but they are not the same point.

General Pseudometric Definition

A general pseudo metric is a function that is semidefinite, symmetric and subadditive. Its codomain is an ordered field. Addition and inequalities are evaluated in the ordered field.

\begin{align*}
        (M0) & \hspace{1cm} \textcolor{lightblue}{d}: X \times X \to \textcolor{Red}{\digamma} \qquad \forall \ x,y,z \in X \\ \\
        (M1) & \hspace{1cm} \textcolor{lightblue}{d}(x,x) = 0                 & (semidefinite)  \\ \\
        (M2) & \hspace{1cm} \textcolor{lightblue}{d}(x,y) = \textcolor{lightblue}{d}(y,x)    & (symmetric) \\ \\
        (M3) & \hspace{1cm} \textcolor{lightblue}{d}(x,y) \leq \textcolor{lightblue}{d}(x,z) + \textcolor{lightblue}{d}(z,y)     & (triangle-ineqaulity)     \\
    \end{align*}

Pseudometric Definition

A pseudo metric is a function that is semidefinite, symmetric and subadditive. Its codomain is the real numbers. Addition and inequalities are evaluated in the real numbers.

\begin{align*}
        (M0) & \hspace{1cm} \textcolor{lightblue}{d}: X \times X \to \textcolor{Red}{\mathbb{R}} \qquad \forall \ x,y,z \in X \\ \\
        (M1) & \hspace{1cm} \textcolor{lightblue}{d}(x,x) = 0                 & (semidefinite)  \\ \\
        (M2) & \hspace{1cm} \textcolor{lightblue}{d}(x,y) = \textcolor{lightblue}{d}(y,x)    & (symmetric) \\ \\
        (M3) & \hspace{1cm} \textcolor{lightblue}{d}(x,y) \leq \textcolor{lightblue}{d}(x,z) + \textcolor{lightblue}{d}(z,y)     & (triangle-ineqaulity)     \\
    \end{align*}