The Mathematics of Distances – Metric Spaces

A metric space is a set with a distance function d(x,y) satisfying: non‑negative, zero only if x=y, symmetric (d(x,y)=d(y,x)), and triangle inequality (d(x,z) ≤ d(x,y)+d(y,z)). The Euclidean plane is a metric space. The taxicab metric (Manhattan distance) is d = |x₁–x₂|+|y₁–y₂|. The discrete metric d(x,y)=1 if x≠y, 0 otherwise. Metrics define convergence, continuity, and compactness – the foundation of real analysis. They allow us to study abstract spaces (function spaces, sequence spaces) that look nothing like ordinary geometry.