Metric spaces

A metric space is a set \(X\) equipped with a symmetric function \(d: X \times X \to [0,\infty)\) that is zero on the diagonal, strictly positive off the diagonal, and satisfies the triangle inequality. Metric spaces are both a fundamental, classical concept in mathematics and a topic of active research, especially in metric geometry.

Sometimes one or more of the following axioms are too restrictive:

  1. Finiteness: \(d(x,y) < \infty\)
  2. Symmetry: \(d(x,y) = d(y,x)\)
  3. Separation: \(x = y\) whenever \(d(x,y) = 0\)

When all three axioms are dropped, the result is a Lawvere metric space: a set \(X\) equipped with a function \(d: X \times X \to [0,\infty]\) that is zero on the diagonal and satisfies the triangle inequality. Such a structure is a category enriched in the extended, nonnegative real numbers.

Literature

Metric spaces belong to the standard undergraduate curriculum and are therefore treated in nearly every introductory textbook on real analysis or topology. For example:

  • Carothers, 2000: Real analysis (doi), Part I: Metric spaces
  • Sternberg & Loomis, 1990: Advanced calculus, Ch. 4: Compactness and completeness
  • Munkres, 2000: Topology, 2nd ed., Sec. 20: The metric topology & Sec. 21: The metric topology (continued)

For more advanced material, see the references on metric geometry.

New metrics from old

The question of how new metric spaces can be derived from old is exhaustively, and exhaustingly, explored by the literature on “metric-preserving functions”.

  • Doboš, 1998: Metric preserving functions (pdf)
  • Corazza, 1999: Introduction to metric-preserving functions (doi, pdf)
  • Matkowski & Świątkowski, 1993: Subadditive functions and partial converses of Minkowski’s and Mulholland’s inequalities (eudml , pdf)