Metric space

A metric space is a set equipped with a distance function that assigns each pair of points a nonnegative distance, with distance zero only between identical points, symmetry, and the triangle inequality.

Metric: A function assigning nonnegative distances to pairs of points, with identity, symmetry, and the triangle inequality. This function supplies the defining structure of every metric space.
Open ball: The set of points whose distance from a given center is less than a specified radius. Open balls define neighborhoods using the space's distance function.
Banach fixed-point theorem: A theorem guaranteeing a unique fixed point for a contraction on a complete metric space. Completeness and distance contraction together guarantee convergence to a solution.
Uniform space: A set with a structure specifying which pairs of points are uniformly close, without requiring numerical distances. It retains uniform notions of closeness while generalizing metric spaces.
Triangle inequality: An inequality stating that one side of a triangle is no longer than the sum of the other two. It constrains distances so detours cannot be shorter than direct separation.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another as the indices increase. Its definition uses distances between sequence terms, not a presumed limit.
Hausdorff distance: A distance between subsets of a metric space, based on how far each set lies from the other. It extends point-to-point distance to compare shapes and sets.
Topological space: A set equipped with a collection of open subsets satisfying axioms for continuity and neighborhood structure. A metric induces a topology, but topology forgets specific distance values.
Euclidean distance: The usual straight-line distance between points in Euclidean space. It is the standard geometric metric and a familiar model for the axioms.
Convergence in a metric space: A sequence converges to a point when its distances from that point tend to zero. The metric gives a precise test for approaching a limit.

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