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An écart for a set $X$ is a non-negative real-valued function $e:X\times X\rightarrow \mathbb{R}$ such that

  1. $e(x,y)=0$ if and only if $x=y$;

  2. for each positive number $s$ there is a positive number $r$ such that $e(x,z)<s$ whenever $e(x,y)$ and $e(y,z)$ are both less than $r$.

My question is how to construct a non-negative function $d:X\times X\rightarrow \mathbb{R}$ such that

  1. $d(x,y)=0$ if and only if $x=y$;
  2. $d(x,y)+d(y,z)\geq d(x,z)$ for all $x,y$ and $z$ in $X$;
  3. for each positive number $s$ there is a positive number $r$ such that $d(x,y)<s$ whenever $e(x,y)<r$ and similarly $e(x,y)<s$ whenever $d(x,y)<r$.

My attempt: Intuitively，I define a function $d$ as $d(x,y)=\inf\{e(x,u)+e(u,y):u\in X\}$ and the first requirement above of $d$ can be easily proved. However, I stuck on the second requirement. I was wondering that this construction of $d$ is right ? Can someone give me some hints? Thanks!

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Jay
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1 Answers1

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Intuitively, I define a function $d$ as $d(x,y)=\inf\{e(x,u)+e(u,y):u\in X\}$ and the first requirement above of $d$ can be easily proved. However, I stuck on the second requirement. I was wondering that this construction of $d$ is right ? Can someone give me some hints?

The standard construction is

$$d(x,y)=\inf\{e(x_0,x_1)+\dots +e(x_{n-1},x_n):x_0,\dots, x_n\in X, x_0=x, x_n=y\}.$$

Alex Ravsky
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