Elementary aspects of the geometry of metric spaces

The setting of metric spaces is very natural for numerous questions concerning manifolds, norms, and fractal sets, and a few of the main ingredients are surveyed here.

💡 Research Summary

The paper provides a comprehensive survey of the elementary geometry of metric spaces, emphasizing how this simple yet powerful framework underlies many areas of modern mathematics. It begins by recalling the definition of a metric space (X, d) and the four axioms that a distance function must satisfy: non‑negativity, identity of indiscernibles, symmetry, and the triangle inequality. Classic examples such as the real line, Euclidean spaces, discrete metrics, and more sophisticated infinite‑dimensional instances—including the uniform metric on C(