example of pseudometric space

Let $X={ℝ}^2$ and consider the function $d:X\times X$ to the non-negative real numbers given by

$d((x_1,x_2),(y_1,y_2))=|x_1-y_1|.$

Then $d(x,x)=|x_1-x_1|=0$, $d(x,y)=|x_1-y_1|=|y_1-x_1|=d(y,z)$ and the triangle inequality follows from the triangle inequality on ${ℝ}^1$, so $(X,d)$ satisfies the defining conditions of a pseudometric space.

Note, however, that this is not an example of a metric space, since we can have two distinct points that are distance 0 from each other, e.g.

$d((2,3),(2,5))=|2-2|=0.$

Other examples:

• Let $X$ be a set, $x_0\in X$, and let $F(X)$ be functions $X\to R$.Then $d(f,g)=|f(x_0)-g(x_0)|$is a pseudometric on $F(X)$ [1].

• If $X$ is a vector space and $p$ is a seminorm over $X$, then$d(x,y)=p(x-y)$ is a pseudometric on $X$.

• The trivial pseudometric $d(x,y)=0$ for all $x,y\in X$ is apseudometric.