4
$\begingroup$

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.

$\endgroup$

1 Answer 1

8
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.