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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).
For the purposes of defining compact sets in a metric space, it's fine to just consider covers by open balls. (if you don't know what a metric space is, think about the plane or 3d space with the usual notion of distance. An open ball is just the set of points that are at a distance less than R from some point). For example, a square in the plane - including its interior points *and* its boundary - is a compact set.