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Suppose we would like to define a metric on New York City, let t : NYC × NYC → R+ is a function that measures the time it takes to travel between two points in New York City. Why doesn’t t satisfy the criteria of being a metric?

I know that a metric must satisfy triangle inequality, non-negativity, and symmetry. I feel like this satisfies all of these conditions. Why is it not a metric, though? Thanks for the help!

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  • $\begingroup$ Do you mean travel by car? $\endgroup$
    – Arthur
    Jul 22 '20 at 23:12
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First and most important, it’s not well-defined: the time needed to get from one point in New York City to another depends on the time, date, and means of transportation. Secondly, it need not be symmetric even at a particular time and date using a particular means of transportation.

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