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A metric space is an ordered pair $(M,d)$, where $M$ is a set and $d:M\times M\rightarrow \mathbb{R}$ is a metric on $M$ such that for any $x,y,z\in M$

  1. $d(x,y)=0\Longleftrightarrow x=y$
  2. $d(x,y)\geq 0$
  3. $d(x,y)=d(y,x)$
  4. $d(x,z)\leq d(x,y) + d(y,z)$

Source: https://en.wikipedia.org/wiki/Metric_space

There are plenty of examples of what are distances functions.

What are some real life examples of a function $d$ where some of these criteria are satisfied and others are not so that it has some of the properties of a distance metric but not all of them? For example, a situation where 2 is not satisfied but some of the others are.

For example, the Jaccard index (aka Tanimoto index) used in QSAR/QSPR studies violates condition 4 based on what I read but satisfies some of the others.

Stan Shunpike
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    What do you mean real life? anyway, $d(f,g) =\int^1_0|f(x)-g(x)|,dx$ is a psedometric. in fails to satisfy (1). One can make a metric space out of this by some set theoretic trick (identification). – Mittens May 13 '21 at 22:02
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    You might look up seminorms. – peter a g May 13 '21 at 22:09

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