# What do we call a metric that doesn't satisfy triangle inequality?

According to wikipedia a metric is a function $d(x,y)$ that satisfies

• non-negativity
• identity of indiscernibles
• symmetry
• triangle inequality

What should we call $d(x,y)$ if the triangle inequality (and probably identity of indiscernibles) doesn't hold? For example $d(x,y)=x^2+y^2-3xy$?

• A “mec”? ($\mathrm{metric} - \mathrm{tri}$) – user1892304 Oct 20 '16 at 15:36
• @user1892304 good job! – dontloo Oct 20 '16 at 15:39
• @G.Sassatelli: Semimetrizable spaces are characterized by having a base $\{B(x,n):\langle x,n\rangle\in X\times\Bbb N\}$ such that $y\in B(x,n)$ iff $x\in B(y,n)$; it’s been many years since I dealt with such things, and I no longer remember any specific examples, but I seem to recall that this combination of first countability and symmetry is enough to substitute for metrizability in some theorems. – Brian M. Scott Oct 20 '16 at 15:50

• Not just Wikipedia: that’s the term that I learned some forty-five years ago, and it’s the term used in Willard’s General Topology (Exercise $23$F). – Brian M. Scott Oct 20 '16 at 15:45