Let (X,d) be a metric space. d is a metric. 1) Is it possible that d(1,2)=d(1,8)? 2) Is it possible that d(1,3)>d(1,100)? If the answer is yes, wouldn't it be weird? The distance between 1 and 3 is larger than the distance between 1 and 100? This is highly counter-intuitive to me... 3) Is it possible that d(1,3)+d(3,7)≠d(1,7)? I am very used to the usual Euclidean distance/metric, in which the above are all impossible. I'm still not entirely comfortable with the idea of a metric space. My playing around with different metrics seem to suggest that the above are possible, but it doesn't seem to make sense to me... May someone explain this? Thanks for any help!