Are there any "spaces" that violate symmetry of metric spaces?
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.
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.
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.