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Definition: A metric space is a nonempty set M of objects (called points) together with a function d from MxM to R (called the metric of the space) satisfying the following four properties for all points x, y, z in M: 1) d(x, x)=0. 2) d(x, y)>0 if x does not equal y. 3) d(x, y)=d(y, x). 4) d(x, y)<= d(x, z) + d(z, y). The metric is the function d.
Source: Mathematical Analysis, second edition by Tom M. Apostol