The metric axioms:
1. d(x,x)=0. You don't have to walk, to get to where you already are.
2. d(x,y)=d(y,x). You could walk over to your mate, but he could just as easily walk over to you.
3. d(a,c) <= d(a,b)+d(b,c). You don't make your trip any shorter, by taking a detour.
Courtesy of my undergrad. topology lecturer.