Distance

A metric space is a set with a distance function that satisfies three axioms: the triangle inequality, symmetry, and distinguishability of points. Examples of metric spaces include familiar ones such as \(\mathbb {Q}\) and \(\mathbb {R}^n\) . The fundamental concepts of metric spaces are introduced in this chapter, namely, balls, interior and boundary points, open and closed subsets, limit points, and dense subsets. Their important properties are proved, including that any union, and any finite intersection, of open subsets is open.