In the definition of a metric space, replace the triangle inequality by the weaker inequality


*d* (*x*, *z*) ≤ *C* max {*d* (*x*, *y*), *d* (*y*, *z*)},


where C is a positive constant (depending on the "metric", but not on the points *x*, *y*, *z*). Had structures like this ever been studied? 


One can associate a more or less natural topology to this "metric", calling a set *open* if every point belonging to the set has a ball of positive radius, centered at this point and contained in the set. But I cannot say much on this topology. For instance, it is not obvious (and may be not true) that a ball is an open set. Neither could I prove that this topology is Hausdorff. 


Any information, reference, etc. would be welcome. 



weak metric space
Yes, they were introduced in valuation theory by Emil Artin and remain present in many contemporary treatments, including mine: see


[http://alpha.math.uga.edu/~pete/8410Chapter1.pdf](http://alpha.math.uga.edu/%7Epete/8410Chapter1.pdf)


especially Section 1.2 and


[http://alpha.math.uga.edu/~pete/8410Chapter2.pdf](http://alpha.math.uga.edu/%7Epete/8410Chapter2.pdf)


[**Added**: as RW has justly pointed out, my answer here makes sense in the context of valuation theory only. The procedure that I give from passing from a "weak metric" to a metric is not going to work in general, I think, but only in the presence of some additional algebraic structure. If I were the OP, I might not choose this as the accepted answer, or at least not yet.]


The basic idea here is to consider two such guys equivalent if one can be obtained from the other by the operation of raising $d$ to some positive real number power $\alpha$. In such a way, one can make the constant $C \leq 2$ in which case one gets an actual metric. The topology one gets in this way is easily seen to be independent of the choice of $\alpha$.


As it happens, when writing up these notes for a course I taught last spring I also thought a little bit about trying to define "balls" with respect to such a weak metric (i.e., without first renormalizing). I didn't get anywhere with this either.


Finally, I should say that in valuation theory at least, these "weak metrics" (in the valuation theoretic context I called them "Artin absolute values" and then after the course was over decided to change to the terminology to just "absolute values", while retaining the word "norm" for such a guy which was actually a metric) come up as a useful tool but are not really studied on their own or in any deep way: I have yet to see a text on valuation theory where weak norms appear after page 20 or so. Whether they may have wider applicability in some other context, I couldn't say...


