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The concept of a distance, or metric, between the elements of a set is well defined in topology. Let B be any set with x, y, z Є B, and let D be a function from the Cartesian product, B X B, into the set of real numbers, R. D is called a metric on B if the following four statements hold:

1) D(x,y) ≥ 0 for all x, y

2) D(x,y) = D(y,x) for all x, y

3) D(x,y) = 0 if and only if x = yfor all x, y

4) D(x,y) + D(y,z) ≥ D(x,z) for all x, y, z

In this case, the set B having metric D is called a metric space, and is often notated as B, D.

For more information and related definitions, see the links below.

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Q: How do you define a metric space through the use of sets?
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