# example of pseudometric space

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

 $\displaystyle d((x_{1},x_{2}),(y_{1},y_{2}))=|x_{1}-y_{1}|.$

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 $\mathbb{R}^{1}$, 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.

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

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 a pseudometric.

## References

• 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
Title example of pseudometric space ExampleOfPseudometricSpace 2013-03-22 14:40:24 2013-03-22 14:40:24 mathcam (2727) mathcam (2727) 6 mathcam (2727) Example msc 54E35 Seminorm VectorSpace MetricSpace Metric trivial pseudometric