Geodesic
Weakly metric spaces
Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part 12, Tome 352 (2008), pp. 94-105 
A. A. Ivanov. Weakly metric spaces. Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part 12, Tome 352 (2008), pp. 94-105. http://geodesic.mathdoc.fr/item/ZNSL_2008_352_a1/
@article{ZNSL_2008_352_a1,
     author = {A. A. Ivanov},
     title = {Weakly metric spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {94--105},
     year = {2008},
     volume = {352},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_352_a1/}
}
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The paper contains some initial results on the theory of weakly metric spaces. The weak triangle axiom: for any $\varepsilon>0$ there exists $\delta>0$ such that for any points $x,y$, and $z$ with $d(y,z)\leq\delta$ the inequality $d(x,z)\leq d(x,y)+\varepsilon$ holds.