Symmetry of a distance metric for a generating set of Topology

Question

I was trying to prove that $\epsilon$-balls defined based on the shortest travel-time distance in a transportation network is a valid generating set for a topology of points on a transportation network.

For this I used the distance-to-itself-is-zero and triangle inequality properties of shortest travel-time distance.

My question is why do we lay emphasis on distances being symmetric (and metric in general) to be able to generate a topology (euclidean distance etc.) when it does not seem to be needed in the proofs.

Answer 1

Non-symmetric distances are called quasi-metrics and they often are used in topology and topological algebra, (e.g., for studying paratopological groups, see http://arxiv.org/abs/1412.2239). For generating a topology it suffices to have only one axiom $d(x,x)=0$. Distances satisfying this unique axiom are called premetrics. By Proposition 3.7 in http://arxiv.org/abs/0901.0236 , the topology of a topological space $X$ is generated by some premetric if and only if $X$ is weakly first-countable in the sense of Arhangelski. For a non-trivial application of premetrics (to contructing non-separably connected spaces), see http://arxiv.org/abs/0901.0236