Let $A, B$, and $C$ be three points in a metric space. Suppose $AB=5, BC=7,$ and $AC=10$ Is it possible that these points lie on one line of this metric space?
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2There is no notion of a line in a metric space... What do you mean? – Mushu Nrek Aug 28 '20 at 18:38
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1Lines are not defined in a generic metric space. – Michael Hoppe Aug 28 '20 at 18:39
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What is supposed to mean $AB$? – azif00 Aug 28 '20 at 19:40
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I presume $AB$ is shorthand for $d(A, B)$. For the first two commenters, you can give the following definition: three points $a, b, c$ in a metric space are collinear if the triangle inequality is an equality, that is,
$$d(a, b) + d(b, c) = d(a, c)$$
and in this case we say that $b$ is between $a$ and $c$. With this definition the answer to the OP's question is clear. Jelly, if you're using a different definition it would be good to indicate so. You can see these definitions, for example, in this paper.
Qiaochu Yuan
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