Why does one necessarily need the triangle inequality

Question

I'm studying basic topological, metric and normed spaces and I am curious why one of the axioms of both a metric and a norm is the triangle inequality. It makes some sense to me having the triangle inequality satisfied sometimes, but I don't quite understand why you necessarily need it for a general metric/norm, since there are still topologies arising from it. Could someone maybe elaborate a little bit?

Answer 1

In a metric space, the triangle inequality says that if you go from point $x$ to point $y$ via point $z$, the distance is at least as big as the distance from $x$ to $y$. I find it hard to argue against that.

Besides the above philosophical fact, negating the the triangle inequality in a normed space is basically negating continuity of the sum: if $$\|a+b\|>\|a\|+\|b\|$$ for all $a$ in a subspace, then $$\|ta+b\|>\|ta\|+\|b\|$$ for all scalars $t$, and so if the sum is continuous we would the contradiction $\|b\|>\|b\|$.

All in all, we usually want triangles where the sum of two sides is greater than the other side, and where the sum in our normed space is continuous. So we postulate the triangle inequality, which by the way is satisfied by scores of natural norms and distances.

Answer 2

One difficulty without the triangle inequality is that the set of open balls may fail to be a base for a topology, so there may be little,if any, relation between the "metric" and some usual, useful, topologies on the space.