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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 $d(x,y)=x^2+y^2-3xy$?

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    $\begingroup$ A “mec”? ($\mathrm{metric} - \mathrm{tri}$) $\endgroup$ – user1892304 Oct 20 '16 at 15:36
  • $\begingroup$ @user1892304 good job! $\endgroup$ – dontloo Oct 20 '16 at 15:39
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    $\begingroup$ @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. $\endgroup$ – Brian M. Scott Oct 20 '16 at 15:50
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It is called a semimetric according to Wikipedia.

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  • $\begingroup$ 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). $\endgroup$ – Brian M. Scott Oct 20 '16 at 15:45

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