"This is a clear explanation, but I think some f..."

The first requirement effectively states both that the distance from an element to itself is 0, and that the distance between non\-identical elements must be greater than 0\. The second requirement asserts that a metric must be commutative; informally the distance from $~$a$~$ to $~$b$~$ must be the same as the distance from $~$b$~$ to $~$a$~$\. Finally, the third requirement is known as the triangle inequality and asserts that the distance from $~$a$~$ to $~$c$~$ is at most as large as the sum of the distances from $~$a$~$ to $~$b$~$ and from $~$b$~$ to $~$c$~$\. It is named as such because in Euclidean space, the points $~$a$~$, $~$b$~$, and $~$c$~$ form a triangle, and the inequality requires that the length of one side of the triangle is not longer than the sum of the lengths of the other two sides; violating this would mean that the shortest path between two points is no longer the straight line between them\.

This is a clear explanation, but I think some formatting changes could enable readers to grok it even more quickly.

Suppose a reader understands two of the three requirements and just needs an explanation of the third. It would be cool if they could find the sentences they're looking for w/o having to scan a whole paragraph looking for the words, "first", "second", or "third".

I think we can achieve this by A) moving each explanation right under the equation / inequality it's talking about, or B) putting the three explanations in a second numbered list, or C) leaving the three explanations in a paragraph, but use the numerals 1, 2, and 3 within the paragraph. Might require some experimentation to see what looks best.