Metric Spaces – Are There Spaces That Violate Symmetry?

metric-spaces

While reading about metric spaces, the following question struck me. We know the following definition of pseudometric spaces and metric spaces:

Suppose $d: X \times X \rightarrow \mathbb{R}$ and that for all $x,y,z \in X$:

$1. d(x,y) \geq 0$

$2. d(x,x)=0$

$3. d(x,y)=d(y,x)\space\space\space\space\space$ (Symmetry)

$4. d(x,z) \leq d(x,y)+d(y,z)$ (Triangle Inequality)

Such a "distance function" $d$ is called a pseudometric on X. The
pair $(X,d)$ is called a pseudometric space.

If $d$ satisfies:

$5.$ when $x \neq y,$ then $d(x,y)>0$,

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

Now, $\ell_2^2$ with $d: \ell_2^2 \times \ell_2^2 \rightarrow \mathbb{R}$ violates the property of triangle inequality. Any pseudometric space $(X,d)$ would violate the non-negativity of metric spaces, since they have at least two points $x \neq y$ for which $d(x,y)=0$.

Similarly, are there any "spaces" that violate symmetry of metric spaces? If not, how do we justify mathematically?

Thank you in advance.

Best Answer

There are two closely related classes of asymmetric metric spaces that come to mind, although they are not something you would encounter until, say, an upper level graduate course on low dimensional geometric topology. Namely:

  1. The Teichmuller space of a surface equipped with Thurston's asymmetric log Lipschitz metric;
  2. The outer space of a free group equipped with the asymmetric log Lipschitz metric.
Related Question