Definition and Examples

Real Analysis · Axiom Academy

A set plus a distance function obeying four axioms — the minimum a "distance" needs for analysis to work anywhere. 1. The Definition and Its Four Axioms A metric space is a pair (X, d) where X is a set and assigns a real number to every pair of points. To deserve the name distance , d must satisfy four axioms for all . Axiom 2 — Identity of indiscernibles The distance is zero exactly when the two points coincide — distinct points are always a positive distance apart. The distance from x to y equals the distance from y to x . Going straight is never longer than detouring through a third point y . 2. The Real Line: d(x,y) = |x - y| The simplest metric space is with the absolute-value distance . The four axioms are exactly the familiar properties of — and the triangle inequality here is the absolute-value inequality . distance on = length of the gap between the points Why the triangle inequality holds Take x = 1 , an intermediate y = 6 , and z = 4 . The direct gap is |x - z| = 3 . The detour through y costs |x - y| + |y - z| = 5 + 2 = 7 . Indeed : passing through y overshoots z and has to double back, so it can only be longer. 3. The Plane: the Euclidean Metric On the everyday distance is the straight-line length given by the Pythagorean theorem — the Euclidean metric d_2 . It extends to coordinate-by-coordinate. root of the sum of the squared coordinate gaps For x = (1, 2) and y = (4, 6) the coordinate gaps are and , so

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