TITLE: How can I visualize the discrete metric?
QUESTION [6 upvotes]: I saw that in a real analysis proof, they used a proof by contradiction where the metric was a discrete metric. That is, distance is defined to be 1 if the points ARENT the same and 0 if the points are the same. I was trying to visualize how this would look for a unit circle. Does such a visualization exist? What is the correct way of viewing it in 2 dimensions? Thank you!

REPLY [9 votes]: Think of a collection of cities. Whenever you want to travel from one city to another, you have to use a strange and special train service. The train system is designed so that, no matter which two cities you travel between, the journey always takes one hour. Thinking of it another way ... no matter which city you're in, every other city is just one hour away.
In this scenario, the unit circle is the set of all cities, except the one you're currently in.
Thinking of "distance" as "travel time" is not unusual. If you ask how far it is from A to B, people will often say something like "it's about 20 minutes".
There is a related question here.