Mathematics - Real Analysis - Metric spaces - Generalizing Distance

10 steps: (1) A metric space (X,d) is a set X with a distance function d:X×X→ℝ. (2) Axioms: d(x,y)≥0, d(x,y)=0⟺x=y, d(x,y)=d(y,x), triangle inequality d(x,z)≤d(x,y)+d(y,z). (3) Example: ℝⁿ with Euclidean distance. (4) Example: discrete metric d(x,y)=1 if x≠y, 0 if x=y. (5) Example: function space C[