Unneeded Restriction in Metric Spaces? 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?
 A: Generality has a price: the fewer axioms we impose, the less we can prove. Certainly we could impose no axioms, but then what kind of a theory would that be? The triangle inequality has the benefit of holding in a lot of examples that we care about, so things we can prove about metrics are widely useful. 
In fact, I would argue that the triangle inequality is by far the most important axiom in the definition of a metric space - see Motivation for triangle inequality for details - and that you should think about dropping all of the others before the triangle inequality!
A: Intuively, I think that the triangle states the $\mathrm d(x,y)$ is the "shortest path" from $x$ to $y$. For instance, consider $x,y$ in $\mathbb{R}^2$, and suppose that you can only move through straight lines. Then, no matter which point $z$ in you choose, you always get
\begin{equation}
\mathrm d(x,y) \le \mathrm d(x,z) + \mathrm d(z,y).
\end{equation}
That means that the shorter way is going directly from $x$ to $y$, you can't make a better path choosing some point $z$ to pass before you get to $y$ -- supposing your starting point is $x$.
As someone here mentioned, it's also really important, since we use it to prove several important results. For example, we use it to prove that metric spaces are Hausdorff, which implies the uniqueness of limits -- a very basic result that one would expect of a metric space theory.
