Are there any "spaces" that violate symmetry of 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.
 A: 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: 


*

*The Teichmuller space of a surface equipped with Thurston's asymmetric log Lipschitz metric; 

*The outer space of a free group equipped with the asymmetric log Lipschitz metric. 

A: There is an entire area of geometry which deals with asymmetric metrics, it is called Finsler geometry. A Finsler metric on a smooth (connected) manifold $M$ is given by a choice of a function $F$ on tangent spaces of $M$ satisfying certain restrictions. Then the Finsler distance between points $p, q$ in $M$ is defined as 
$$
d_F(p,q)=\inf_c \int_{0}^{1} F(c'(t))dt,
$$
where the infimum is taken over all paths $c: [0,1]\to M$ connecting $p$ to $q$. This Finsler distance function satisfies all the axioms of a metric except for the symmetry.  
A Finsler metric is reversible if $F(-v)=F(v)$. Reversible metrics result in symmetric distance functions. 
A: The Kullback Leibler divergence between two probability distributions tells you how much information samples from one distribution give you to reject the assumption that the samples come actually from the other distribution.  There is no reason this should be symmetric and indeed it isn't.  The other assumptions of a metric are satisfied though. 
A: Metric spaces where distances are in $[0,\infty]$ and which drop symmetry but still satisfy the triangle inequality and and $d(x,x)\ge 0$ are called generalised metric spaces in the influential paper by Lawvere http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html
A nice example of a space where asymmetry is important is given by finite or infinite words over an alphabet, see Rutten http://www.cwi.nl/~janr/papers/files-of-papers/1996-tcs170.pdf . The idea is that $d(w,v)=0$ means that the word $w$ is a prefix of the word $v$ and that $d(w,v)=2^{-n}$ if the first letter where $w$ differs from $v$ is at position $n+1$. 
Ok, this is an ultrametric space. But as the question was about asymmetry, it is still a nice example. And I think that metric space examples can be constructed along the same lines.
