I understand that the idea of the metric developed from a generalization of the idea of the Euclidean metric, wherein the triangle inequality holds. However, why does this actually need to be a definitional necessity for a metric? Would it not be more beneficial, in the name of generality, to remove this qualification?
Unneeded Restriction in Metric Spaces?
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4There is the concept of [semimetric](http://en.wikipedia.org/wiki/Quasi-metric#Semimetrics), which is precisely a function that satisfies the first three axioms of a metric, but not the triangle inequality. There are also quasimetrics (drop symmetry) and pseudometrics (allow $d(x,y)=0$ with $x\neq y$). – 2011-11-26
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5To echo Qiaochu: the more generality, usually the less you can say. Compare the theory of [groups](http://en.wikipedia.org/wiki/Group_%28mathematics%29) with the theory of [magmas](http://en.wikipedia.org/wiki/Magma_%28mathematics%29). Also, you want to get the generality "just right": the concept of metric turns out to be flexible enough to include a lot of interesting examples, but rigid enough to allow you to prove a lot of interesting things. – 2011-11-26