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Definition. A metric space is a set X together with a function d (called a metric or distance function) which assigns a real number d (x, y) to every pair x, y X satisfying the properties (or axioms): 1  d (x, y) 0 and d (x, y) = 0 x = y, 2  d (x, y) = d (y, x), 3  d (x, y) + d (y, z) d (x, z).
In mathematics, space is an unbounded continuum (unbroken set of points) in which exactly three numerical coordinates are necessary to uniquely define the location of any particular point. It is sometimes called 3-space because it contains three distance dimensions. If a continuum requires fewer or more than three coordinates (dimensions) to uniquely define the location of a point, that continuum is sometimes called n-space or n-dimensional space, where n is the number of dimensions.