metric space

A metric space is simply a collection of points together with a rule for measuring distances between them. The moment you can say how far apart two things are, an enormous amount of analysis follows: you can speak of points being close, of sequences converging, of sets being open or closed, all without ever mentioning coordinates or formulas for area.

Formally a metric space is a pair (X, d) where X is a set and d is a metric on X. The same underlying set can be turned into different metric spaces by choosing different metrics, so strictly speaking the metric is part of the data, not an afterthought. Points of X need not be numbers — they can be functions, sequences, matrices, or geometric figures, as long as a sensible distance is defined.

Metric spaces are the natural home for the epsilon arguments of calculus, generalized far beyond the real line. The familiar statement that x_n converges to L if for every epsilon > 0 eventually |x_n - L| < epsilon becomes, verbatim, the definition of convergence with |x_n - L| replaced by d(x_n, L). This is why metric spaces are the bridge from ordinary calculus to abstract analysis.