Function that's a metric on one space but not another?

Question

Is there a function which makes sense on two sets and is a metric on one but not the other? I can't seem to come up with an example or a proof a metric on one set implies it is on every other one it can be defined.

Thanks

Answer 1

Such issues are encountered when one attempts to glue metric spaces together, even when the parts to be glued are disjoint. For example, let $A$ be the interval $[0,1]$ with the standard metric $d_A(x,y)=|x-y|$, and $B=[2,3]$ with the metric $d_B(x,y) = 10|x-y|$. We could try to put these together into one space $X=[0,1]\cup [2,3] $, extending the metric as $d(x,y)=|x-y|$ when $x\in A$, $y\in B$ or vice versa. The resulting function $d:X\times X\to \mathbb R$ is not a metric because, e.g., $d(2,3)>d(1,2)+d(1,3)$. The restrictions of $d$ to $A\times A$ and $B\times B$ are metrics, however.