I'm having a hard time understanding what the definition of a metric is. From what I think I understand, it's just a method of measurement between $2$ points in $\mathbb R^n$? Is that somewhere along the lines correct?
Then I have a homework problem that says: Consider the distance function $d:M\times M\to\mathbb R$ and then prove that $d$ is continuous with respect to the natural sum metric defined on $M\times M$, namely $d_{sum}((p,q),(p',q'))=d(p,p')+d(q,q')$.
I just don't understand what it is I'm supposed to prove. On top of that metric spaces seem so foreign and strange to me that I just don't know how to wrap my head around it.
Best Answer
A metric is supposed to quantify distance. Most common type is the so called Euclidean distance that you know as the hypotenuse.
Now consider a cab driver who charges by the miles. That is distance too, but in a non-conventional geometry. He does not drill through buildings, he is restricted to available roads. So you have a "taxi-cab" metric, for the simple case where all roads are perpendicular. On a grid a meaningful distance is then $d((a,b),(x,y))=|a-x|+|b-y|$ which might be associated with $L_1$ norms. Your GPS device also takes you through a road that is shortest but not straight in the conventional sense.
Most important property of a metric is the triangle inequality. (You just won't be happy if the cab does not go through the shortest path.) So in order to go from $A$ to $C$ his chosen path better not be longer than any trip from $A$ to $B$ and then $B$ to $C$.