Here's the question :
Suppose C is a circle and for each a,b belonging to C, define d(a,b) to be the distance along the line segment form a to b. Then d is a metric on C.
Now i don't really understand the meaning of this statement :
The fact that the route from a to b goes outside C is irrelevant ; a metric is a simply a function defined on ordered pairs and does not take into account any "route traveled" from one point to another.
Are'nt we suppose to define the metric within the given set, which in this case would be the circle C ? Then why do we consider a route that is outside the set, at all? Also does this mean that we cannot define the distance from a to b as the minor arc or the major arc between them , because route does not matter ?
I am a little confused. Please Help.